ArturoMartinezJimene

Transcription

A Grand Canonical Monte Carlo Molecular Study
of a Weak Polyampholyte
Thesis by
Arturo Martinez Jimenez
In Partial Fulfillment of the Requirements
For the Degree of
Master of Science in Earth Science and
Engineering
King Abdullah University of Science and Technology, Thuwal,
Kingdom of Saudi Arabia
(May, 2016)
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The thesis of Arturo Martinez Jimenez is approved by the examination committee
Committee Chairperson: Shuyu Sun
Committee Member: Ibrahim Hoteit
Committee Member: Omar Knio
Committee Member: Arun Kumar Narayanan Nair
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Copyright ©2016
All Rights Reserved
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ABSTRACT
A Grand Canonical Monte Carlo Molecular Study of a Weak
Polyampholyte
Over the last few decades, there has been an increasing interest in the study of
charged polymers for applications such as desalination of water, flocculation, sewage
treatment, and enhanced oil recovery. Polyelectrolyte chains containing both positively and negatively charged units (polyampholytes) have been recently studied as
viscosity-control agents in enhanced oil recovery, and as entrapping macromolecules
for protection and delayed release of enzymes in hydraulic fracturing. In this study we
performed Monte Carlo molecular simulations in a grand canonical ensemble to study
the behavior of a weak polyampholyte in a dilute regime. Weak polyampholytes have
the ability to dissociate in a limited pH, which makes them interesting for applications that require a pH-triggerable response. The titration behaviors of diblock and
random polyampholytes are simulated as a function of solvent quality, electrostatic
strength, and salt concentration. For diblock polyampholyte chains in hydrophobic
solvents, transition between tadpole-like and globule conformation occurs with variations in the solution pH. Random polyampholytes present extended, globule, and
pearl-necklace conformations at different solvent conditions and pH values. At high
ionic strength, electrostatic interactions in the polyampholytes become screened and
the chains are mostly in globule state.
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ACKNOWLEDGEMENTS
I want to express my gratitude to the King Abdullah University of Science and Technology for the education received and the trust deposited on me. Thank you KAUST
for allowing me to be part of this incredible multicultural community. I want to thank
Dr. Shuyu Sun for the opportunity he gave me to be part of his research group during
my stay in the university. Special thanks to Dr. Arun Kumar Narayanan Nair for all
the support and the knowledge he provided me for this project.
I dedicate this thesis to my family; especially to my parents for always encourage
me to pursue my goals. Thank you Virginia Jimenez Treviño and Arturo Martinez
Villegas, without your support I probably wouldn’t be where I am now. My gratitude
also goes to my friends Pablo de la Garza, Aldo Banderas and Gilberto Mendoza for
being there for me during my stay in this university. Thank you guys for all the
support you provided me. Finally, this thesis is also dedicated to my great friend
Daniel Eduardo Elizondo because he has always been encouraging me to continue
pursuing my goals regardless of the distance.
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TABLE OF CONTENTS
Examination Committee Approval
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Copyright
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Abstract
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Acknowledgements
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List of Abbreviations
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List of Figures
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1 Introduction
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2 Research Motivation and Literature Review
2.1 Existing related work on polyampholytes . . . . . . . . . . . . . . . .
2.2 Charged polymers in oil recovery processes . . . . . . . . . . . . . . .
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3 Model Description
3.1 Motivation . . . . . . . . . . . . . . .
3.2 Bead-spring model . . . . . . . . . .
3.3 Chemical properties of the system . .
3.3.1 Polyampholyte in the presence
3.4 Monte Carlo algorithm . . . . . . . .
3.4.1 Trial moves . . . . . . . . . .
3.4.2 Algorithm description . . . .
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4 Results
4.1 Diblock polyampholyte in the absence of salt . . . .
4.2 Diblock polyampholyte in a high salt concentration
4.3 Random polyampholyte in the absence of salt . . .
4.4 Random polyampholyte in a high salt concentration
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5 Concluding Remarks
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Future Research Work . . . . . . . . . . . . . . . . . . . . . . . . . .
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References
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Appendices
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LIST OF ABBREVIATIONS
FENE
Finite extensible nonlinear elastic
L-J
Lennard-Jones
MC
MD
Monte Carlo
Molecular Dynamics
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LIST OF FIGURES
3.1
3.2
3.3
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
Example of a simulated polyampholyte. Green, blue and red beads
correspond to neutral, negative and positive charged monomers. . . .
Trial moves attempted during the simulation (3D simulations were carried out but 2D movements are shown for convenience of illustration).
Charge fraction of both chains during different Monte Carlo cycles for
λB = 1, LJ = 0.5 and pH = 9 in the absence of salt . . . . . . . . . .
Titration curves of the diblock polyampholyte in the absence of electrostatic interactions (theory) and for λB = 0.25 with cs ≈ 10−1 mol/L
(simulation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Titration curves for block B at different LJ and λB values in the absence of salt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conformational behavior of a weak diblock polyampholyte for λB = 1
Radius of gyration of block B for λB = 1 and different LJ values . . .
Conformational behavior of a weak diblock polyampholyte for λB = 0.5
Radius of gyration of block B for λB = 0.5 and different LJ values . .
Radius of gyration of block B for λB = 0.25 and different LJ values .
Titration curves for block B at different LJ and λB values in the presence of a high salt concentration . . . . . . . . . . . . . . . . . . . . .
Conformational behavior of a weak diblock polyampholyte for λB = 1
and λD = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Radius of gyration of block B for λB = 1 and λD = 1 at different LJ
values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
and λD = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Radius of gyration of block B for λB = 0.5 and λD = 1 at different LJ
values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.14 Conformational behavior of a weak diblock polyampholyte for λB =
0.25 and λD = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.15 Radius of gyration of block B for λB = 0.25 and λD = 1 at different
LJ values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.16 Conformational behavior of a weak random polyampholyte for λB = 1
in the absence of salt . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.17 Titration curves for type B monomers at different LJ and λB values
in the absence of salt . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.18 Conformational behavior of a weak random polyampholyte for λB =
0.5 in the absence of salt . . . . . . . . . . . . . . . . . . . . . . . . .
0.25 in the absence of salt . . . . . . . . . . . . . . . . . . . . . . . .
4.20 Radius of gyration for LJ = 0.5 at different λB values . . . . . . . . .
4.21 Titration curves for type B monomers at different LJ and λB values
in the presence of a high salt concentration . . . . . . . . . . . . . . .
4.22 Conformational behavior of a weak random polyampholyte for λB = 1
and λD = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.5 and λD = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.25 and λD = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.25 Radius of gyration for LJ = 0.5 at different λB values in the presence
of a high salt concentration . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 1
Introduction
Charged polymers, also known as polyelectrolytes, have attracted much attention during the past two decades due to their unique properties and great range of applications.
One of their main characteristics is the ability to dissociate charges in polar solvents,
such as water, an ability that many polymers do not posses because of their hydrocarbon backbone[1]. Based on the dissociation behavior, these macromolecules can
be catalogued into strong or weak polyelectrolytes. Strong polyelectrolytes present a
frozen configuration of the charges along the chain; their distribution depends only
on the initial chemistry and does not change with large variations in the solution pH.
On the other hand, the number of charged monomers in weak polyelectrolytes varies
as a function of the pH, meaning that the ionization sites along the chain are not
static [2].
A more complex type of charged polymers are those composed of monomers having
both positive and negative ionizable groups. These are called polyampholytes and
are present in nature, for example, as proteins [3]. Proteins such as gelatin, bovine
serum albumin, and synthetic copolymers made of monomers with acidic and basic
groups are examples of polyampholytes [4].
Because polyampholytes have a similarity with biological macromolecules in nature, they have awakened an interest in fields such as biology, materials science and
soft matter research [5, 6]. In the present, polyampholytes are used for applications
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including desalination of water, sewage treatment, flocculation, coagulation, drilling
fluids, enhanced oil recovery, among others [7]. In the petroleum field, polyampholytes
have been studied as possible viscosity-control agents in enhanced petroleum recovery
and polyelectrolytes as entrapping agents for enzymes required during the hydraulic
fracturing of oil wells [8, 9, 10].
Therefore, there is a strong motivation to try to understand and determine the
behavior of weak polyelectrolytes and polyampholytes for pH-triggerable applications.
However, the combination of the polymer properties and electrostatic interactions
between their charged groups results in a complex behavior difficult to understand
theoretically.
A great number of studies have been performed to investigate the behavior of
polyelectrolytes under different conditions. Monte Carlo (MC) molecular simulations
and Molecular Dynamics (MD) simulations have been extensively used to perform
research in the polymer field. Strong and weak polyelectrolytes have been studied
for different parameters such as solvent quality, salt concentrations, and electrostatic
interactions strength over the last two decades.
In this research, we performed a MC simulation in a grand canonical ensemble
to study the conformational properties, titration curves and radius of gyration of a
weak polyampholyte under poor solvent conditions. Two different structures of the
polyampholyte were investigated; a diblock and a random one, and they were studied
for an extensive pH range. Salt concentration in the solvent and different electrostatic
interaction strengths were also considered in the investigation.
The remainder of the thesis is organized as follows. In the second chapter, a
literature review of different studies that have been made over the last two decades,
in the charged polymers field, is presented. Chapter three introduces the molecular
simulation method used for this research and describes the model followed to simulate
our polyampholyte chain. It contains information about the different conditions to
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which our polyampholyte was subject, and the way in which they were tuned during
our simulations. Chapter four presents the results of the research for the different
polyampholytes simulated. The conformational properties of weak polyampholytes,
titration curves and radius of gyration are discussed and analyzed in detail. Finally,
chapter five presents the final conclusions of this work and establishes the future of
the research.
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Chapter 2
Research Motivation and
Literature Review
2.1
Existing related work on polyampholytes
Over the last three decades, a great amount of theoretical and experimental work has
been performed to understand the behavior of charged polymers. These polymers
contain ionizable groups that dissociate under certain conditions, leaving charged
chains and counterions in the solution [4]. As mentioned before, charged polymers
can be polyelectrolytes if they only contain a single sign of charged monomers or
polyampholytes if they contain both on the same chain [6]. Proteins, for example,
are one of the many charged polymers found in nature. Because of their applications in desalination of water, flocculation, oil recovery processes, materials science,
soft matter research among other, polyelectrolytes and polyampholytes have received
considerable interest [6, 11].
These polymers not only can be classified depending on the sign of their charges,
but also based on their dissociation behavior. In the chemistry community, the terms
weak and strong polyelectrolytes are widely used, while in the physics community
they are referred as annealed and quenched polyelectrolytes [6]. When the charges
along the polyelectrolyte do not vary with large changes in the pH, the charges are
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said to be in a frozen configuration and it only depends on the initial chemistry [2].
In these cases, the charged polymer presents a quenched configuration. When the
amount of charged sites can be tuned by changing the pH and therefore ionization
sites are able to move along the chain, we have an annealed charged polymer [6, 12].
Molecular simulation methods have been used in the past to study the behavior of
both polyelectrolytes and polyampholytes. MC molecular simulations and MD simulations are two of the most used procedures to study the behavior of charged polymers
under different conditions. In these simulations, there are several parameters that can
be tuned in order to obtain a different response of the systems studied. Parameters
can change in order to study the effect of the solvent quality [6, 13, 2, 14, 15, 1] or
the effect of salt concentration in the solvent [12, 16, 17].
MC simulations were performed in 1992 by Sassi et al. to study titration and
configurational properties of weak polyelectrolytes under poor solvent conditions [13].
Titration curves show the charge fraction of the polyelectrolyte as a function of the
solution pH and they are useful in the understanding of the acid-base properties
of annealed charged polymers [6]. Using a cubic lattice, Sassi et al. performed
simulations in the grand canonical ensemble to determine the effect of the electrostatic
interactions on the ionization of a weak polyelectrolyte. The number of potentially
ionizable groups was fixed, along with their intrinsic dissociation constant and the
solution pH. By fixing these last two parameters, ionization of the chain is allowed to
find its equilibrium value, and titration curves are obtained by performing simulations
over a range of pH [13].
Stevens and Kremer, in 1995, performed MD simulations of linear polyelectrolytes
and presented a fundamental model for polyelectrolytes in solution. Their system was
not only composed of charged monomers, but also included counterions, and they determined that counterion condensation plays an important role in the chain structure.
Simulation results were in function of density and chain length, presenting information
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about the structures of polyelectrolytes in salt free solutions [18]. A year later, Ullner
and Jönson performed MC simulations for a rigid rod polyelectrolyte and a flexible
one in the presence of salt. Bond length between monomers was fixed for both chains
and the charges interacted via a screened Coulomb potential. Salt concentrations
of 0.001, 0.01 and 0.1 mol/L were studied for chains with 80-1000 monomers. Salt
concentration, chain length, chain ionization and bond length between monomers
were tuned in order to determine the conformational properties and the apparent
dissociation constant of polyelectrolytes [17].
Micka, Holm and Kremer performed another MD study in 1999. Flexible polyelectrolytes under poor solvent conditions were simulated without the presence of
salt. Structural properties of the chains and the solvent were reported for strongly
charged quenched polyelectrolytes. Two main parameters were varied in this study,
the polymer density in the solvent and the electrostatic interaction strength. Results
showed a transition from a dominating electrostatic interaction regime to a regime of
strong screening in which the hydrophobic interactions dominate [14].
In 2003, Limbach and Holm, using MD studied the polyelectrolyte properties under poor solvent conditions in a dilute regime. Their system consisted of a quenched
polyelectrolyte under conditions in which the interaction between chains was neglected, while the counterion concentration remained finite. Polyelectrolyte conformations were investigated varying different parameters such as chain length, Coulomb
interaction strength, charge fraction and hydrophobicity of the chain, in order to
determine in which regime the competition between the hydrophobic interactions
and the Coulomb interactions lead to the formation of the so called pearl-necklace
structures[1]. The same year, Chang and Yethiraj using also MD, studied the behavior
of charged flexible polymer molecules under poor solvent conditions in the absence of
salt. Polymer molecules were simulated as chains of charged spheres, counterions were
also considered as charged spheres and the solvent molecules were explicitly modeled
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as uncharged spheres. The response of the system was studied under different solvent
conditions and polymer concentrations [15].
Ulrich, Laguecir and Stoll using MC simulations studied annealed hydrophobic
polyelectrolytes in 2005. A grand canonical ensemble was used to determine polyelectrolyte conformations and titration curves in the presence of screened Coulomb
interactions. Different type of conformations such as extended, pearl-necklaces and
collapsed structures were obtained under different ionic strengths and hydrophobic
interactions. The charge fraction in the polyelectrolyte chain was studied for different
values of pH in order to obtain the titration curves. Their results showed that hydrophobic interactions play an important role in the formation of compact structures
by decreasing the charge fraction for a given pH value [2].
Polyampholyte behavior has also been the subject of study in recent years, however, because of the presence of both negative and positive charges along the chain,
monomer interactions result more complex than those in the polyelectrolyte cases.
Jeon and Dobrynin studied the polyampholyte behavior in the presence of a polyelectrolyte chain using MC simulations. Chains were simulated using a bead-spring
model, where single monomers are considered as Lennard-Jones (L-J) particles. The
polyelectrolyte used was completely charged, and the polyampholyte was overall neutral with the same amount of positive and negative charges. Two different charge
arrangements were used for the polyampholyte, a random one and an alternating
one. The formation of complexes between both chains was studied for different electrostatic interaction strengths, and in the case of the alternating polyampholyte,
different block sizes were considered. Results showed that complexes formed between
a random polyampholyte and a polyelectrolyte are stronger than the case were an
alternating polyampholyte is used [4].
Same authors performed in 2005 a similar study to determine the effect of the solvent quality and salt concentration in the formation of polyelectrolyte-polyampholyte
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complexes. In their study, the polyelectrolyte chain consisted of 187 beads and every third bead a negative charge was present in the chain. In the polyampholyte, all
beads in the chain were positively or negatively charged with and overall charge equal
to zero, for a random charge distribution and a diblock charge sequence along the
chain. Explicit counterions were considered to maintain the electroneutrality of the
system due to the charges in the polyampholyte and polyelectrolyte chains, and salt
concentration was accounted by adding salt ions at different concentrations. Poor
solvent conditions were considered for the polyelectrolyte chain while the polyampholyte was in a theta solvent. Results concluded that polyelectrolytes in a poor
solvent tend to form necklace-like structures, caused by the presence of two different mechanisms that lead to their formation. In the presence of weak electrostatic
interactions, the necklace formations are due to the optimization of the short-range
interactions between monomers and the repulsion of the charged monomers in the
backbone chain. For strong electrostatic interactions the necklace formation result
from counterion condensation; while short-range interactions still play an important
role, the attraction between charged monomers and counterions and the electrostatic
repulsion between uncompensated charges contribute significantly to the stability of
the necklace structures. About the complexes formation, their results showed that in
the presence of both, a random and a diblock polyampholyte, polyelectrolyte chains
change their necklace structure to a collapsed globule, which according to them is the
result of neutralization in the polyelectrolyte charge by the polyampholyte [16].
Ulrich, Seijo and Stoll performed grand canonical MC simulations with a screened
Coulomb potential in 2007 to study the behavior of weak polyampholytes. The effect
of stiffness, structure of the polyampholyte and ionic concentration in the solvent,
on the titration curves and chain conformations were investigated. Stiffness effect
was studied by comparing a rod-like rigid polyampholyte and a flexible one. Comparison between different primary structures was performed for the case of diblock,
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octablock, alternating and random polyampholytes. Their results showed that the
primary structure of the polyampholyte plays an important role in the acid-base properties and in the charge distribution for the rod-like polyampholytes. On the other
hand, flexible polyampholytes favor attractive electrostatic interactions which lead to
more compact conformation and different acid-base properties than the ones present
in the rod-like structures [19].
In 2014 Nair, Uyaver and Sun performed the study of a weak polyelectrolyte and
a diblock weak polyampholyte under good and poor solvent conditions. Both chains
were in contact with a reservoir of constant chemical potential given by the solution
pH. Two oppositely charged weak polyelectrolyte blocks, with each block containing the same amount of monomers, formed the diblock polyampholyte. Titration
curves and conformational properties were obtained at different pH values. For the
polyampholyte under poor solvent conditions, the results revealed a discrete transition between an extended and a collapsed conformation by varying the solution pH.
In the good solvent case, the polyampholyte does not present discrete transitions, and
results suggest that for an average charge fraction of one of the blocks smaller than
0.4 and equal to one, the block behaves as a single polyelectrolyte [6].
2.2
Charged polymers in oil recovery processes
Over the last couple of decades, a great number of polyampholytes have been synthesized, some of which have showed potential in applications where there is a high
salinity in the media such as enhanced oil recovery, non-biofouling coatings and mechanically tough hydrogels [20]. One of the interesting properties of polyampholytes
its their anti-polyelectrolyte behavior in polar solvents. While polyelectrolyte collapses in the presence of salt, polyampholyte present a higher solubility and increase
the solution viscosity [20]. This effect favors the use of polyampholytes in high elec-
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trolyte solutions. They can be used as thickeners in brine solutions, in flocculation
or in oil recovery processes [11].
Polyampholytes have been studied in recent years for oil recovery applications.
Enhanced oil recovery has been one of the fields where polyampholyte implementation has been considered. Because the easily recoverable oil is running out and a great
amount of oil remains in the reservoirs after conventional methods have been used,
implementation of enhanced oil recovery suggest an alternative method to guarantee
a continuing production [21]. Polyelectrolytes have been previously used for enhanced
oil recovery applications, especially to modify the rheological properties of the displacing fluid. The use of this type of polymers has showed an improvement in the
water-oil mobility ratio, however, their effectiveness decreases in the presence of high
salt concentrations and high temperatures[21]. On the other hand, at dilute regimes
different polyampholytes exhibit unique globule to coil transitions with increasing
salt concentrations, leading to subsequent increases in the solution viscosity [8]. For
polyampholytes composed of weak groups, changes in the solution pH can lead to
changes in the conformational behavior of the chain. This type of polyampholytes is
useful for applications where a pH-triggerable response is needed.
In 2007, Ezell and McCormick studied the pH and salt responsive behavior of solutions containing different polyampholytes. Their study suggested that the aqueous
solution properties have a dependence on the groups that conform the polyampholyte.
They tested their solution for polyampholytes with variations in their anionic groups.
Their results showed that at low salt concentrations, viscosity of the solution is governed by electrostatic interactions. By contrast, electrostatic interactions are eliminated at high salt concentrations because of charge screening, making the viscosity
dependent of the polyampholyte composition [8]. Maia et al. compared, in 2008,
the viscosity response of a conventional polyacrylamide polymer, extensively used in
polymer flooding, and a hydrophobically modified polyacrylamide. The performance
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of the polymer solutions was tested in a porous medium through core flood experiments in Botucatu sandstone. Their experiments showed that the hydrophobically
modified polyacrylamide, at low concentrations, increases the viscosity of the solution
in the presence of salt to higher values than the regular polyacrylamide, suggesting
that these type of polymers might increase the oil recovery in high salinity reservoirs
[22].
In 2011, Barati et al. studied the use of polyampholytes as entrapping agents for
enzymes required in the hydraulic fracturing of production wells. Their investigation
states that water-based polymer gels are used in the oil industry to increase the
viscosity of the fluids used in the hydraulic fracturing of production wells. These fluids
form filter cakes in the faces of the fractures that must be degraded once the fracturing
process is done. Degradation of the polymer gels and the filter cake is achieved with
the use of enzymes in order to attain a high conductivity during production. The
existing techniques add the enzymes directly to the gel or in capsules that are crushed
when the fractures close. In the first case the gel may be degraded prematurely
leading to an inefficient fracture propagation, while in the second one, a nonuniformly
degradation of the gel may be obtained leading to a reduced hydraulic conductivity.
In order to obtain a delayed release and a uniformly degradation, polyampholytes
were used to entrap the enzymes. Their results suggest that the combination of
the uniform distribution and the delayed release of the enzymes in the polymer gel,
showed promise for improved cleanup after the fracturing process [9].
The same year, Zou et al. performed a study where anionic and cationic acrylamide polymers where modified and tested for enhanced oil recovery purposes. The
polymers were evaluated on aspects such as intrinsic viscosity, interfacial tension and
stability experiments. Their results demonstrated that the modified polymers show
better performances on interfacial tension, salt resistance, temperature tolerance, viscosification property and shear resistance than the usual polyacrylamide used in the
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recovery processes [23]. In 2012, Barati et al. performed another study on polyampholytes as coating agents for enzymes required during hydraulic fracturing. This
time, polyampholyte complexes were used to entrap two different enzymes used in
the oil industry to degrade the polymer gels after fracturing. They not only studied the delay in the gel degradation that polyampholytes cause by entrapping the
enzymes, but also the protection that the coating gives to the enzyme at elevated
temperatures and pH values [10].
The previous studies prove that polyampholytes have been recently considered for
applications in the oil industry, where an efficient recovery is always desired. For that
reason, studies about the conformational behavior of polyampholytes, which have
shown an effect on the viscosity of the injected fluids, are still needed in order to
consider the different conditions found in the reservoirs.
In this research, MC simulations in the grand canonical ensemble were performed
for a diblock polyampholyte and a random polyampholyte in a dilute regime. The
chains were studied under poor solvent conditions for different electrostatic interaction
strengths and a high salt concentration. The results obtained were used to create
titration curves and conformational diagrams. In the following section the model used
to simulate the polyampholyte chain, as well as the MC algorithm, are explained in
detail. The results obtained from the simulations are then analyzed and compared
for both polyampholyte structures.
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Chapter 3
Model Description
3.1
Motivation
As mentioned before, polyampholytes are macromolecules possessing the ability to
ionize in polar solvents. Like polyelectrolytes, they are also catalogued in two groups
depending on the ionization degree of their monomers. The difference between these
two previously mentioned macromolecules lies in their composition. While polyelectrolytes can only be either a polyacid or a polybase, polyampholytes contain both
groups. Our interest is the molecular study of a polyampholyte formed by two bonded
polyelectrolyte chains, one of them being a polyacid and the other one a polybase,
and a polyampholyte with acid and base monomers randomly distributed along the
chain. The molecular study is carried out in the absence of polyelectrolyte counterions
and is extended to include a high salt concentration. Short range monomer-monomer
interactions and long range electrostatic interactions between charged monomers can
lead to a complex behavior of the system, as compared with a solution of neutral
polymers [24, 25, 26].
The first step for a simulation study on polyelectrolytes is to select the model
that will represent the main characteristics that describe the physical and chemical
phenomena that interest us. In order to optimize the computational calculations, the
model selected should be as simple as possible, but still capable of accounting for
24
the important molecular interactions [25]. Because polyampholytes are just a conformation of two different polyelectrolytes, same assumption can be made to represent
them. Figure 3.1 shows how a polyampholyte is represented in our simulations. The
monomers are simulated as hard spheres and they can carry a neutral, positive or
negative charge depending on the monomer type they represent. In order to understand and study the conformational behavior of polyampholytes, a simple model is
used to represent the structures and interactions within the solvent.
Figure 3.1: Example of a simulated polyampholyte. Green, blue and red beads correspond to neutral, negative and positive charged monomers.
3.2
Bead-spring model
According to Underhill and Doyle, the wide range of time and length scales in a
polymer system difficult the modeling. Because of this, the idea of coarse-graining
has been a recurrent theme in the development of polymer models [27]. In a coarsegrain model, monomers in a polyelectrolyte chain are represented as linked charged
beads and the solvent as a constant dielectric medium [25, 6]. The ions in the system
are typically considered as rigid spheres, but in our case, we are working in a regime
were counterions do not play a significant role [1, 6]. As mentioned by Underhill
and Doyle: “The goal in coarse-graining is to produce a model that has reduced
complexity such that it is tractable to calculate the properties of the model while
25
simultaneously capturing molecular properties to sufficient accuracy”[27].
One of the main properties in polymers, that distinguish them from small molecules,
is the existence of an elastic restoring force in the presence of small deformations. The
presence of this restoring force must be present in the model, and it usually is represented by springs, making a bead-spring representation a suitable choice for a polymer
model [27]. The polyampholyte simulated in our research is represented by the widely
used standard bead-spring chain model [6, 14, 18, 1].
The diblock polyampholyte chain is composed of block A, which is a polyacid with
monomers carrying a zero or a negative charge depending if they are in a protonated
or deprotonated state, and block B that represents a polybase carrying a positive
charge in the protonated state and a zero charge in the deprotonated state [6]. In the
random polyampholyte, polyacid and polybase monomers are randomly distributed
along the chain and they can be in the same state as in the diblock chain. The same
number of polyacid and polybase monomers is present in both polyampholyte chains,
in this case making the diblock chain a symmetric polymer. Monomers in the chain
interact via a truncated-shifted L-J potential[14, 1, 15, 4, 6]:
ULJ (r) =

h

4LJ ( σ )12 − ( σ )6 − (
r


r
σ 12
)
Rc
+ ( Rσc )6
i
; r ≤ Rc
(3.1)
0 ; r ≥ Rc
where r is the distance between two monomers and Rc is a cutoff radius after which
the potential has a zero value. The parameter σ establishes the bead diameter while
LJ controls the strength of the short-range interactions [14, 4, 6]. Polyampholytes
have been studied in either good or poor solvent conditions [14, 6]. Under poor
solvent conditions, a cutoff radius Rc = 2.5σ has been used in the literature to give
the monomers a short-range attraction, while LJ values higher than 0.34kB T , which
represents the theta point for neutral polymers, are used to study the dependence on
the solvent quality [1]. The theta point for neutral polymers represents the value at
26
which attractive and repulsive interactions cancel up, thus leaving the polymer with
an ideal random walk behavior [14]. For this reason we are interested in values higher
than the theta point, where interactions between the polymers play an important role
in their behavior.
To represent the connectivity of the monomers along the chain an interaction potential must be used. Usually, the connectivity between the monomers is represented
by a harmonic potential [25]:
Ubond (r) = kbond (r − R0 )2
(3.2)
where kbond represents the spring constant for the potential, r =| ri+1 − ri | is the
distance between two consecutive monomers and R0 represents an equilibrium bond
length [25]. However, the electrostatic interactions between monomers can cause a
strong strain enough to extend the bond to distances that are not physically realistic.
For this reason, a potential that limits the extension of the bond distance and creates
stiffer bond lengths is typically selected [25, 14]. The finite extensible nonlinear
elastic (FENE) potential is one of the most used models in the polymer literature
[18, 14, 1, 4, 15, 16, 6]:
r2
1
2
UF EN E (r) = − kF EN E R0 ln 1 − 2
2
R0
(3.3)
where the spring constant kF EN E is 7kB T /σ 2 and the equilibrium bond length 2σ,
same values than the ones used by several polyelectrolyte studies [4, 18, 16, 6]. The
equilibrium bond length sets the maximum allowed extension for the bond. Finally,
the Coulomb potential gives the electrostatic interaction between charged monomers:
UC (rij ) = kB T
λB qi qj
rij
(3.4)
where rij represents the distance between the centers of two charged monomers, qi and
27
qj the valence of the monomers charge and λB = e2 /(ε0 εkB T ) is the Bjerrum length.
The Bjerrum length is the value that determines the strength of the electrostatic
interaction, and is defined as the distance at which the thermal and electrostatic energy have the same value [14]. As mentioned before, to avoid counterion interactions,
Bjerrum length values were set to lower values than σ [6]. The solvent is taken into
account via the dielectric constant ε in the Bjerrum length.
3.3
Chemical properties of the system
We do not model the solvent explicitly in our simulations. Instead, several considerations must be done in order to account for the chemical properties of the system. In
an aqueous solution with the presence of hydrogen ions, the chemical equilibrium of a
charged polymer is calculated by the intrinsic dissociation of the monomers [19]. Our
chain is composed of two monomer types: type A monomers represent a weak acidic
site that can be neutral or negatively charged, while type B monomers represent a
weak base that can either be neutral or positively charged [3]. Dissociation of an acid
monomer in an aqueous solution is expressed as [2]:
HA H+ + A−
(3.5)
where HA represents an undissociated acid, H+ a dissociated hydrogen and A− a
dissociated acid. The law of mass action establishes that the dissociation constant at
which the concentrations of dissociated hydrogen ions and dissociated acids does not
change is represented as:
H+ A−
KA =
[HA]
(3.6)
On the other hand, the dissociation of a base monomer is represented as [3]:
28
BH+ B + H+
(3.7)
[B] H+
KB = + BH
(3.8)
with dissociation constant:
In general, both values KA and KB , will depend on the charge fraction in their
respective polyelectrolytes [13]:



[A − ]
;
+[HA]
f= [ ] +

 [BH ]+ ;
[B]+[BH ]
A−
(acid)
(3.9)
(base)
This is a result of the electrostatic interactions between charged monomers. Because a change in the charge fraction will result in a change in the dissociation constant
of the monomer groups, it is useful to determine an intrinsic dissociation constant K0
for each group [13]. The role of this intrinsic dissociation constant is to characterize
a monomer group in the polymer independently of the electrostatic interactions [28].
However, in the literature, a more practical term is the negative logarithm to base
ten of the intrinsic dissociation constants [13]:


 pK A = −log10 (K A )
0
0

pK B = −log (K B )
0
10
(3.10)
0
For the MC algorithm, the polyampholyte chain is immersed in a reservoir of
charges with a fixed chemical potential determined by the pH of the solution and the
intrinsic dissociation values of the monomers: [19, 3, 6]:


-ln10(pH − pK A ) ; (acid)
µ
0
=
 ln10(pH − pK B ) ; (base)
kB T

0
(3.11)
29
3.3.1
Polyampholyte in the presence of salt
In our research, we include the presence of a high salt concentration in our simulations.
When salt is added, charged monomers electrostatic interactions become short-ranged
[12]. According to Netz: “the interactions between charges are screened due to the
presence of mobile ions in the surrounding space”[29]. Screened interactions are
modeled via a Debye-Hückel potential instead of the normal Coulomb potential [30].
Debye-Hückel potential is defined as:
UDH (rij ) = kB T
λB qi qj −rij /λD
e
rij
(3.12)
where λD is the Debye length, which set the length at which electrostatic interactions persist. The Debye length used in the Debye-Hückel potential is related to the
Bjerrum length by the formula [30]:
λD = (8πλB cs )−1/2
(3.13)
where cs represents the salt concentration in mol/L. In the presence of screened
interactions, the self-energy of a released proton contributes to the chemical potential,
and equations (3.11) become [12]:


-ln10(pH − pK A ) − λB /λD ;
0
µ
=

kB T
 ln10(pH − pK0B ) − λB /λD ;
3.4
(acid)
(3.14)
(base)
Monte Carlo algorithm
Our algorithm computes the equilibrium conditions for a polyampholyte chain in a
dilute regime. It starts by generating the initial positions of the different monomers
along the chain and calculating the initial energy of the chain. Then, as a result
of different random trial moves in the chain, a large number of configurations are
30
generated. According to the MC acceptance conditions, not all the random moves
are accepted. In the case of our grand canonical ensemble, the acceptance condition
is established as:
e−∆E/kB T ≥ random number between 0 and 1
(3.15)
The energy change value ∆E is defined as:
∆E = ∆Ec ± µ
(3.16)
where the ∆Ec value, represents the change in the configurational energy defined as
[6]:
∆Ec = ∆ULJ + ∆UF EN E + ∆UC
(3.17)
The sign in equation (3.13) depends on whether a protonation or deprotonation
was attempted, and the µ value depends on the monomer type.
3.4.1
Trial moves
MC simulations were performed using the Metropolis algorithm, in which a great
number of successive trial chain conformations are generated to obtain a sampling
of low-energy conformations [2]. Three trial moves were performed during the MC
cycles: local, pivot and charge movements. In every cycle, a number of local and
charge movements equal to the number of monomers in the chain are performed
in randomly selected monomers. A pivot movements is attempted every 200 local
movements, in accordance with the simulations performed by Nair et al. [6].
For the local movement, a randomly selected particle is moved in a 3D direction
with a random step. The random movement is performed in each one of the 3D directions separately and it is randomly selected from the interval −0.1σ ≤ ∆x, ∆y, ∆z ≤
31
(a) 2D representation of a local movement
(b) Representation of a charge movement
(c) 2D representation of a pivot movement
Figure 3.2: Trial moves attempted during the simulation (3D simulations were carried
out but 2D movements are shown for convenience of illustration).
0.1σ [6]. Because we are simulating a polyampholyte chain in the dilute regime, where
interaction with other chains is not considered, no boundary conditions are required
and the polyampholyte chain is free to move in all directions. Only restriction in the
movement is that the distance between monomers does not exceed the equilibrium
bond length established by the (FENE) potential. If the equilibrium bond length in
one of the connections along the chain is exceeded, the movement is rejected. If the
Metropolis criterion is satisfied, the movement is accepted and the new configuration
is saved. Figure 3.2(a) illustrates a local movement.
In the charge movement, a monomer is randomly selected and its charge value is
switched depending on the monomer type and the current charge. Once the switch is
32
Figure 3.3: Charge fraction of both chains during different Monte Carlo cycles for
λB = 1, LJ = 0.5 and pH = 9 in the absence of salt
done, equation (3.13) is used to determine the change in energy and the acceptance
condition is evaluated. If the acceptance condition is satisfied, the switch is accepted
and the new configuration is saved. Figure 3.2(b) illustrates an example of how a
monomer is switched from one charged state to another. Finally, every 200 local
movement attempts, a pivot movement is performed in the polyampholyte chain. A
randomly selected monomer is chosen as the pivot for a 3D rotation in a randomly
selected angle between zero and 2π. After the rotation is performed the acceptance
condition is evaluated. If the acceptance condition is satisfied, the rotation is accepted
and the new configuration is saved. In our simulations, the polyampholyte is equilibrated for a period of 5 × 106 cycles, and the properties of the chain are obtained by
averaging over 3 × 106 cycles after the equilibration process [6]. Figure 3.3 illustrates
an example of how the charge fraction of both chains of a diblock polyampholyte
varies during the simulation cycles. Sampling of the charge fraction is performed
more frequently in the production process.
33
3.4.2
Algorithm description
1. Get the input values: intrinsic dissociation constants pK0A and pK0B , pH, number of equilibration and production cycles, number of monomers in the chain
Nm , initial charge fraction of the chain, LJ , λB and λD if salt is included.
2. Generate the initial positions for the monomers in the chain and compute its
initial energy.
3. Select a random monomer and perform a local movement on it.
4. Compute the energy of the new configuration and check the acceptance condition. If the movement is accepted, then the configuration of the chain changes
and the new state is saved. If not, the previous configuration is restored.
5. Select a random monomer and perform a charge movement on it.
6. Repeat the step four.
7. If the overall number of local movements attempted is a multiple of 200, a pivot
movement is performed in a random monomer.
8. Repeat the step four.
9. Repeat the steps, from step three to step eight, a number of times equal to the
number of monomers in the chain.
10. Repeat step nine for a number of times equal to the number of equilibration
and production cycles determined in the input values
34
Chapter 4
Results
4.1
Diblock polyampholyte in the absence of salt
MC simulations were performed on different nodes containing HP BL685c Quad sockets with AMD Opteron 6376 processors, 16 cores per socket (2.3GHz) and 256GB
memory on each node. For this project we have investigated weak polyampholytes
consisting of 256 monomers, half of them being weak acids and the other half weak
bases. The absence of explicit counterions in our model is treated by choosing the
Bjerrum length value λB ≤ σ [6]. In our simulations, the LJ unit is kB T and length
unit is σ. The Bjerrum length for water at room temperature is approximately 7.14Å
[6, 16]. Average bond length between monomers is given by b ≈ σ, this also establishes that the Manning parameter, defined as ξM = f λB /b, is below the critical value
ξM = 1 at which counterions interactions have to be considered [14]. Reduced units
are used in our simulations in order to reduce the risk of an overflow or underflow
in our computations. Also, by using reduced units almost all our quantities are of
order one (between 10−3 and 103 ) [31]. The values for pK0A and pK0B were chosen
as 5.5 and 7.5, similar values to the ones reported for the apparent pKa values of
a weak poly(methacrylic acid)/poly[2-(dimethylamino)ethyl methacrylate] polyampholyte [32]. These values can be tuned in order to account for different types of
monomers with different pKa values.
35
Simulations were performed using σ = 1 and kB T = 1 as length and energy units
respectively. Bjerrum length values λB = 1, λB = 0.5 and λB = 0.25 are used under
different LJ values to simulate the polyampholyte chains in a determined pH range.
Usually, attractive interaction in L-J particles is tuned by changing the temperature
of the system [14]. Same as with Micka et al., one purpose of this research is to study
the effect of the poor solvent under constant λB and T . This is the reason of why the
hydrophobicity of the polymer is varied through the LJ as an independent variable
[14]. The energy scale is fixed by setting kB T = 1. Variation of the Bjerrum length
might seem merely as an academic study, however, the important parameter that
gives the strength of the electrostatic interactions is u = λB /b. Hence, varying the
average bond length between monomers, which can be done during the polyampholyte
synthesis, can produce a similar effect as the one obtained by varying the Bjerrum
length [30].
The isoelectric point corresponds to the pH value where the net charge of the
macromolecule is equal to zero and it is defined as [5]:
pI =
pK0A + pK0B
2
(4.1)
Diblock polyampholyte conformations should present symmetric behavior around
the isoelectric point. Confirmation of this can be made through titration curves of
both blocks in the whole pH range. In an ideal case where no electrostatic interactions
are considered, the charge fraction degree of each block is determined by the equations:
f=





1
A −pH
;
(acid)
B
;
(base)
1+10pK0
1
1+10pH−pK0
(4.2)
Figure 4.1 shows the ideal titration curves of our polyampholyte in the absence of
electrostatic interactions, and the titration curves of a chain with λB = 0.25 under a
salt concentration of cs ≈ 10−1 mol/L.
36
Figure 4.1: Titration curves of the diblock polyampholyte in the absence of electrostatic interactions (theory) and for λB = 0.25 with cs ≈ 10−1 mol/L (simulation)
Typically, when the overall charge of the polyampholyte is zero or close to zero,
chains tend to collapse into globules [33]. Hydrophobic interactions also play an
important role in the chain behavior. Unlike single polyelectrolyte studies, no pearlnecklace conformations are present in these diagrams [1]. Conformational diagrams
will be analyzed for three LJ values: LJ = 0.5 which corresponds to a chain that
shows signs of hydrophobicity, LJ = 1.5 for and hydrophobic chain and LJ = 2.5 for
a chain in a deep hydrophobic region [14]. LJ values higher than 2.5 were simulated
in accordance with studies by Ulrich et al. [2]. However, no significant changes are
observed in terms of shape or size when compared to LJ = 2.5 case. Therefore,
further analysis in this region was not performed. To support the analysis of the
conformational behavior of the polyampholyte chain, titration curves for the block
with B type monomers were obtained for the three LJ values at different λB .
To characterize the polyampholyte chain quantitatively during the changes in the
pH, the root-mean-square radius of gyration of block B was also analyzed for the
cases previously mentioned [2, 6]. Radius of gyration is defined as:
37
Figure 4.2: Titration curves for block B at different LJ and λB values in the absence
of salt
Rg2 =
1 X
(ri − rj )2
2
2Nm i,j
(4.3)
where N m represents the number of monomers considered, while ri and rj their
respective positions. This quantity gives a general view on the size of the polymer
chain. Because we know that block A and block B have an overall symmetric behavior
around the isoelectric point, obtaining the radius of one of them is enough to describe
the chain. Also, because at both pH extremes the chain is typically found in a tadpole
conformation (see, e.g., Figure 4.3), calculating the radius of the whole chain will
represent an inaccurate measure. For these two reasons, only results of the radius of
gyration of the B block are presented in this study.
Figure 4.3 shows the conformational behavior of our weak polyampholyte at different pH and LJ values using λB = 1. The highlighted region only shows the behavior
of the polyampholyte for pH values higher than the isoelectric point of the polyampholyte. It is evident that at the isoelectric point of our polyampholyte (pI = 6.5),
collapsed globule like conformations are present for the different LJ values.
38
(a) λB = 1
(b) Highlighted region in (a)
Figure 4.3: Conformational behavior of a weak diblock polyampholyte for λB = 1
For LJ = 0.5, conformational properties of the chain for λB = 1 are illustrated
in Figure 4.3. In all figures, green beads, red beads and blue beads correspond to
neutral, positive and negative charged monomers respectively. It can be noticed
that at the isoelectric point, short-range interactions and electrostatic interactions
between opposite charges dominate causing the polymer to collapse into a globule
like conformation. As seen in Figure 4.2, as the charge asymmetry δf = |f− − f+ |
increases while increasing the pH above the isoelectric point, block B dissociates less
in the solution, causing an overall negative charge in the chain that stretches the
block A.
Size changes in block B can be observed in Figure 4.4. At pH = 1, block B is
completely charged and reaches its maximum elongation value Rg /σ ≈ 30. By the
time pH = 3, block A starts to dissociate and those charges are attracted to the
positive ones in block B, shrinking it to a value of Rg /σ ≈ 16. For the interval
between 4 ≤ pH ≤ 9, block B reaches a size minimum because of the rapid increment
in the charge fraction of block A, causing the collapse of the polyampholyte to a
globule like conformation. After pH ≈ 9, block B suffers two conformational changes.
In the interval 9 / pH / 10, as the pH increases the size of block B does the same.
A reduction in the dissociation of this block is the cause of the elongation because
39
Figure 4.4: Radius of gyration of block B for λB = 1 and different LJ values
the globule like conformation is not stable anymore, causing the attraction of the
charged monomers in both blocks. Finally for pH ' 10, the charge fraction in the
block B keeps decreasing and so its size because of less attractive interactions. This
keeps going until it charge fraction reaches zero and it collapses into a random coil
conformation.
For LJ = 1.5, the results show a different behavior in the polyampholyte chain.
Analyzing the Figure 4.4 for this hydrophobicity, the chain seems to be in a globule
like conformation in almost the whole pH range. Block B only presents an elongation
when its charge degree is approximately one. The abrupt change in the polyampholyte
conformation is caused by the large dissociation in a wider pH range than in the
previous case for LJ = 0.5. As is seen in Figure 4.2, compared to results for LJ = 0.5,
the charge fraction of block B remains above 0.5 for 12 ' pH ' 1. As a result of
increasing the hydrophobicity, in order to reduce its contact surface, more dissociation
is present in the polyampholyte in order to cause the globule like conformation to
become stable in a wider pH range. The fast decay in the charge fraction outside this
range causes both blocks to elongate in their respective extreme, obtaining a tadpole
40
conformation of the polyampholyte for pH ' 12 and pH / 1.
At LJ = 2.5, the chain is in a globule like conformation for the whole pH range.
By increasing the hydrophobicity, short-range interactions are favored. Also, the
range of dissociation for both chains is increased as it is observed in Figure 4.2, where
the charge fraction remains above a value of 0.5 for 13.5 ' pH ' 1 for the case of the
block B. This results in a wider range of stable globule conformation as it is observed
in 4.4, where the radius of gyration of block B does not change with increasing the
charge fraction.
(a) λB = 0.5
Figure 4.5: Conformational behavior of a weak diblock polyampholyte for λB = 0.5
Figure 4.5(a) illustrates the conformational behavior of the polyampholyte chain
for λB = 0.5. For LJ = 0.5, at first sight, the conformations of the chain seem
very similar to the λB = 1 case. However, by analyzing the titration curves and
radius of gyration we can get more information about the influence of a decrease in
the electrostatic interaction strength. As we have mentioned before, by tuning the
λB value we can vary the strength of the electrostatic interactions. Titration curves
suggest that the range of dissociation of block B decreases with a reduction in the
λB value. The charge fraction value of the chain remains over 0.5 for 9 ' pH ' 1
compared to the range for the λB = 1 case which was 10 ' pH ' 1. This reduction
in the electrostatic strength not only has an effect on the dissociation behavior of the
41
chain, but also on its size.
Figure 4.6: Radius of gyration of block B for λB = 0.5 and different LJ values
Figure 4.6 shows that the maximum size achieved by block B is approximately
Rg /σ ≈ 25 and it occurs when it is completely charged. A behavior similar to λB = 1
is present in this case; at high charge fraction values block B presents an extended
conformation. Because block A starts to dissociate at a slightly higher pH value than
in the λB = 1 case, it is expected that the block B maintains its extended state for
a slightly higher pH range. Once dissociation in block A starts, block B shrinks to
achieve a globule like conformation. After this, as pH increases and dissociation in
block B decreases, a similar behavior as in the strong electrostatic strength case is
present: block B starts to extend because of the increase in block A charge fraction
that attracts the few charges in B, and then once the charge fraction in block B
reaches an approximate value of 0.4, it starts to shrink again because of the shortrange interactions. The only difference is that the maximum value achieved in this
process is Rg /σ ≈ 7.5 compared to the Rg /σ ≈ 14 in the λB = 1 case.
For LJ = 1.5, the chain presents a similar behavior to the one for λB = 1. Block
B is in an extended configuration only when the charge fraction is approximately
42
one. In this case, because dissociation of the block A is delayed compared to the
case for λB = 1, this extended configuration remains stable in a slightly bigger pH
range. Also, due to the fact that charge fraction of block A is practically zero when
charge fraction of block B is at its maximum, size of block B in its extended form
at pH = 1 is higher than the size of block B at the same pH for LJ = 1.5 in the
λB = 1 case. The overall behavior of the chain is similar to the case with stronger
electrostatic interactions, where globules like conformations are present for basically
all the different charge fraction values.
For LJ = 2.5, no extended conformations in block B are present even for charge
fraction values very close to one. This was expected as the electrostatic interaction
strength was reduced, and we are in a deep hydrophobic regime where short-range
interactions are favored. However, there is a change in the titration curve for this
case compared to the equivalent case for λB = 1. In the last one, for almost the whole
pH range, charge fraction of the block B remains over a 0.5 value, while for λB = 0.5,
charge fraction remains over a 0.5 value until a pH ≈ 10.5.
(a) λB = 0.25
Figure 4.7(a) shows the conformational behavior of the polyampholyte for an
even lower electrostatic interaction strength. In this case, λB is set to a value of 0.25.
Titration curves for this case at different LJ values are displayed in Figure 4.2 as
43
open symbols. Titration curves at this electrostatic strength show a similar behavior
to the ones for the ideal case where no electrostatic interactions are considered. In
Figure 4.7(b), tadpole like conformations are still present in the polyampholyte chain
in the low hydrophobicity regime.
Figure 4.8: Radius of gyration of block B for λB = 0.25 and different LJ values
By analyzing the titration curves and radius of gyration for block B in Figures 4.2
and 4.8, respectively at LJ = 0.5, we can determine the following: titration curves
evidence the fact that dissociation of block A starts at a higher pH than for the
previous electrostatic strengths. This suggests that tadpole like conformations are
stable in a slightly bigger pH range than in the previous cases. Because electrostatic
interactions are weak, the maximum Rg /σ value for the extended B block is reduced
to a value of approximately 20. Since electrostatic interactions are weak, block B has
a transition from an extended to a globule like conformation as pH increases. No
further changes are present in the overall size of the block B as charge fraction in the
block A increases.
In the hydrophobic region where LJ = 1.5, the extended conformation of block
B is present only when the charge fraction value is very close to one. This indicates
44
that globules are present in the majority of the pH range. Figure 4.2 also shows that
charge fraction for block B decays faster or at lower pH values than for the other cases,
however, because of the weak electrostatic interactions, globule like conformations
are still stable even when charge asymmetry in the chain is high. For LJ = 2.5, the
titration curve of block B presents a similar behavior than for LJ = 1.5, except for
the fact that the charge fraction decays slower after pH ≈ 11. Meaning that block
A has a higher dissociation at pH = 1 than for 1.5 , causing the chains to collapse
into globule like conformations for a bigger pH range. Weak electrostatic and strong
short-range interactions are the cause of this collapse even in the presence of a high
charge asymmetry.
4.2
Diblock polyampholyte in a high salt concentration
Diblock polyampholyte was also studied in the present of a high salt concentration.
Salt screens the electrostatic interactions, making the attractive and repulsive forces
weaker than in pure water [26]. Weakening of the interactions between charged
monomers is caused by the presence of ions in the surrounding space [29]. By using
the Debye-Hückel potential we have been able to introduce a high salt concentration
in our simulations. For our simulations, λD = 1, which for the different λB values
correpond to salt concentrations close to 10−1 mol/L. Polyampholyte conformations
for high salt concentration are studied for the same λB and LJ values than in the
previous section where no salt was considered. Titration curves for the nine different
combinations are contained in Figure 4.9.
Figure 4.9 shows that the dissociation behavior of block B is similar to the ideal
case, where no electrostatic interactions are considered. These results suggest that
the addition of salt plays an important role in the dissociation of the polyampholyte
45
Figure 4.9: Titration curves for block B at different LJ and λB values in the presence
of a high salt concentration
chain. It can be noticed that the pH value at which the dissociation for block B
reaches an equilibrium (fB ≈ 0.5) is around eight for all the cases, which is very close
to the intrinsic dissociation constant selected for type B monomers.
(a) λB = 1
Figure 4.10: Conformational behavior of a weak diblock polyampholyte for λB = 1
and λD = 1
Figure 4.10 contains the conformations for λB = 1. Screening effect is observed
in the polyampholyte chain because most of the diagram shows collapsed globules.
For LJ = 0.5, the blocks show the behavior of an ideal flexible chain only when their
46
charge fraction is close to one. Figure 4.10(b) illustrates better the behavior of both
blocks for pH values above the isoelectric point. At the isoelectric point both blocks
carry the same amount of charge, causing the blocks to collapse into a globule like
shape. As it can be seen in Figure 4.9, after the isoelectric point, charge fraction in
block B decreases rapidly with slightly changes in the pH, while charge fraction in
block A is close to one at this point. Globule conformations remain stable despite
the high charge asymmetry in the block, mainly because electrostatic interactions are
weakened due to the presence of salt. Once the charge asymmetry reaches a critical
point, as it can be seen for pH ' 10, the globule conformation disappear and both
blocks show a different behavior. Because we are in a region where the chain shows
some signs of hydrophobic influence, block B that now is completely neutral collapses,
while block A that is almost fully charged behaves like an ideal polyelectrolyte.
Figure 4.11: Radius of gyration of block B for λB = 1 and λD = 1 at different LJ
values
Figure 4.11 contains the radius of gyration values for the block B for λB = 1 at
different LJ values. It shows that for LJ = 0.5, size of block B has a maximum
of Rg /σ = 10, which is approximately 1/3 the size of the same case but in the
absence of salt. As it can be seen in this figure, the size of block B decays quickly
47
with the minimum change in the charge fraction, reaching a minimum size that does
not change as the charge fraction keeps decreasing. At LJ = 1.5 and LJ = 2.5,
the polyampholyte chain presents a similar behavior in both cases. Globule like
conformations dominate during the whole pH range and the titration curves present
a similar trend. Radius of gyration illustrated in Figure 4.11 confirms the previous
statement, since the size in block B does not change with variations in the charge
fraction. The stability of the globule conformations in the whole pH range, even
for a high charge asymmetry, may be attributed to two factors. The first one is
that both chains are in a region where hydrophobic influence plays a major role in
the conformational behavior, so even in the presence of a high charge asymmetry,
chains tend to reduce their surface area. The second and most important one is
that the screening effect is strong due to the high salt concentration, weakening the
electrostatic interactions and allowing the chains to reduce their surface area.
(a) λB = 0.5
and λD = 1
For the case where the electrostatic interactions have been weakened by reducing
λB = 0.5, Figure 4.12 shows the conformations of the polyampholyte for the different
hydrophobic regions. For LJ = 0.5, images in Figure 4.12(b) reveal some slightly
differences in the configurations compared to the case where λB = 1. The collapsed
48
state found at pH = 6.5 presents a less regular shape. Even though both blocks
present a high charge fraction, because electrostatic interactions have been weakened,
not only by the reduction of λB but by the addition of salt, the collapsed state shows
some irregularities. Block B charge fraction decays faster than for λB = 1 as it can
be seen in Figure 4.9, causing the conformation at pH = 8.0 to be in a collapsed
like configuration instead of a more regular shape. Once block B carries no charges
on it, a similar behavior is observed in the polyampholyte chain: block B collapses
due to the hydrophobic influence and block A behaves like an ideal polyelectrolyte.
However, Figure 4.13 shows that the maximum size of block B is smaller compared
to the case where λB = 1, which means that at its maximum charge fraction value,
block B presents a smaller extension.
Figure 4.13: Radius of gyration of block B for λB = 0.5 and λD = 1 at different LJ
values
For LJ = 1.5 and LJ = 2.5, similar conformations than the ones obtained for
λB = 1 are observed. Titration curves for this cases decay slightly slower than for
LJ = 0.5. The radius of gyration of block B shows that collapsed globules remain
stable even if the charge fraction changes. This means that in these two hydrophobic
regions, decreasing the electrostatic interaction strength only favor the hydrophobic
49
influence, collapsing the chains, making them stables.
Polyampholyte conformations for the cases where λB = 0.25 are showed in Figure
4.14. Collapsed globules are majority in the diagram, and only for the low hydrophobic chain an ideal behavior seems possible. Figure 4.9 illustrates that the titration
curves for the three different LJ values follow the same trend and seem to overlap.
(a) λB = 0.25
and λD = 1
For LJ = 0.5, the conformation of the polyampholyte for a pH = 6.5 doesn’t look
like a regular globule anymore. This is caused by the reduction in the electrostatic
interaction strength, making the attraction between opposite charges weaker. Charge
fraction in block B decays faster than for the two previous λB cases, which means that
at pH = 8, the collapsed state of the chain seems less stable. As pH keeps increasing,
block B becomes neutral and collapses due to the hydrophobic influence, while block
A is completely charged and behaves like and ideal polyelectrolyte. However, because
electrostatic interactions are very weak and screening effect of salt is strong, block
A does not reach the same size as with the previous electrostatic strengths in Figure
4.14(b). Maximum size of block B (which should be similar to the maximum size of
block A) is showed in Figure 4.15. Size of block B differs slightly from the maximum
value obtained for λB = 0.5.
50
Figure 4.15: Radius of gyration of block B for λB = 0.25 and λD = 1 at different LJ
values
For LJ = 1.5 and LJ = 2.5, stable conformations for the polyampholyte chain
keep being globules. The weakness of the electrostatic interactions, the salt screening
effect and the hydrophobic influence cause the chain to collapse even in the presence
of a high charge asymmetry. Titration curves for both LJ show an identical behavior
and the radius of gyration of block B tends to the same value.
4.3
Random polyampholyte in the absence of salt
For a random weak polyampholyte chain, the conformational properties are highly
dependent on the net charge and the charge asymmetry of the chain [24]. A high
charge asymmetry in the chain backbone means the polyampholyte has an excess
of either positive or negative charges. In this case, repulsive interactions dominate
causing the chain to exist in an extended state which might be water soluble [24].
When the number of charged monomers is equal for both monomer types, charge
asymmetry is zero and polyampholyte exist in a globule state that is insoluble in
water [24]. The region between high and zero charge asymmetry has a more complex
51
behavior. Conformational diagrams for a random polyampholyte will be presented
for different λB values, and they will be analyzed for LJ = 0.5. More LJ values
are present in the diagrams, however, because of the presence of pearl-necklace-like
conformations, studies on the size of these chains result more complicated.
Figure 4.16: Conformational behavior of a weak random polyampholyte for λB = 1
in the absence of salt
Figure 4.16 shows the presence of extended, pearl-necklace and globule like configurations. For LJ = 0.5, the chain can go from a globule like state to an extended
one. At the isoelectric point, the presence of a hydrophobic influence and attraction
between opposite charges overcome the repulsion between equal charges, causing the
chain to exist in a collapsed irregular conformation. As the charge asymmetry in the
chain increases, this conformation collapse and the chain starts to stretch because of
the repulsion between charges. Figure 4.17 shows the dissociation behavior of type
B monomers, which is symmetric to the behavior of type A. If we analyze closely
this figure and compare it to Figure 4.2 for the diblock case, it can be noticed that
52
both chains present a very identical dissociation behavior. As a result of this, we
can say that the distribution of the monomers along the chain does not affect their
dissociation behavior.
Figure 4.17: Titration curves for type B monomers at different LJ and λB values in
the absence of salt
According to Figure 4.17, monomer dissociation has a fast decay after a pH ≈ 9.
At pH = 10, we can see that charge fraction of type B monomers is close to 0.5,
while for type A it is almost one. This causes instability on the chain, because of
the repulsive forces between type A monomers, that extend the chain. As charge
asymmetry keeps increasing we should expect a further elongation of the chain. For
LJ = 1.5 the chain exists in globule and pearl-necklace conformations for low and
high charge asymmetries respectively. For type B monomers, the titration curve
shows a slow decay on the charge fraction until a pH ≈ 11.5, meaning that the charge
asymmetry remains low until that value, causing the chain to be in a globule state. It
can be noticed that for a pH = 12, the chain exists in a pearl-necklace state. As the
charge asymmetry in the chain decays, there is a point where the globule becomes
unstable and to reduce its energy it will split into smaller globules separated by a
string of charged monomers. For LJ = 2.5 globule like conformations are stable and
53
the chain does not seem to have a transition to another state. The titration curve of
the type B monomers show that the behavior of the charge fraction is similar to the
LJ = 1.5 case. However, it does not present the fast decay after pH ≈ 11.5, instead
of that it maintains the slope of the decay until an approximate pH value of 13. After
that value the charge asymmetry seems to have a slight jump, which might cause a
pearl-necklace conformation to appear at pH = 14.
Figure 4.18: Conformational behavior of a weak random polyampholyte for λB = 0.5
Figure 4.18 contains the conformations for λB = 0.5. At LJ = 0.5 the chain
exists in collapsed and stretched states. However, the range in which collapsed states
are stables seems to be reduced compared to the case when λB was equal to one.
In the isoelectric point it can be noticed that the chain exists in what it seems to
be a collapsed state rather than a regular globule. Though, as the pH increases (or
decreases), this collapsed state becomes unstable and the chain seems to be in a state
between a globule and an extended chain. The titration curve in Figure 4.17 shows
54
that the charge fraction of type B monomers decays faster than for λB = 1 at the same
LJ value. However, the difference between titration curves for both cases at a pH = 8
seems small, which means that this instability on the chain and the early extension is
caused by the weakening of the electrostatic interactions. As the charge asymmetry
in the chain keeps increasing, the chain becomes more elongated. For LJ = 1.5,
because the chain is in a hydrophobic region and the charge asymmetry of the chain
decays slower around the isoelectric point, globule like conformations remain stable
for a larger pH range. However, as it can be seen in Figure 4.17, after a pH ≈ 10.5 the
charge fraction of type B monomers suffers an abruptly decay. This causes the globule
conformation to collapse at an earlier pH into the pearl-necklace conformation. It is
because of this early collapse that at pH = 12 the pearl-necklace structure seems
more stretched and the globules seem smaller. In the case of LJ = 2.5, the chain is
in a deep hydrophobic region, causing it to be in a globule state for almost the whole
pH range. The titration curve for this case shows a similar trend than the previous
LJ case, however it does not present the abrupt decay until a pH ≈ 12. This might
suggest that for a pH > 12, the chain could adopt a pearl-necklace conformation
because of the high charge asymmetry that the chain present at those pH values.
Last case studied in the absence of salt is for λB = 0.25, which is the weakest
electrostatic strength studied in this research. For LJ = 0.5, Figure 4.19 shows a
globule conformation only when the pH is close to the isoelectric point. The titration
curve in Figure 4.17 reveals that after the isoelectric point the charge fraction of
type B monomers has a fast decay, because by the time pH ≈ 8, the charge fraction
value has decreased approximately a 40%. Charge asymmetry in the chain increases
and for that reason the globule conformation breaks, and a quasi-extended chains
is present at pH ≈ 8. However, because of the weak electrostatic interactions, we
expect that the extension of the chain at this pH does not reach the same value than
for λB = 0.5. Increasing the pH keeps increasing the charge asymmetry causing a
55
further elongation, however, the chain does not reach a full extended state because
of the low λB value. LJ = 1.5 results show that the globules are almost present
in the whole pH range, pearl-necklace-like conformation are observed only for cases
where the charge asymmetry is close to its highest value. The titration curve for this
case in Figure 4.17 shows that the decay of charged type B monomers is slower at
the isoelectric point than for LJ = 0.5, this in addition with the increment in the
hydrophobicity of the chain and the weak electrostatic interaction strength, cause the
globules to be present in a higher pH range. A faster decay in the charge fraction
occurs after pH ≈ 10, increasing the charge asymmetry and making the globules
unstable. This effect causes the presence of a pearl-necklace structure at pH ' 10,
as it can be noticed that this structure appears in Figure 4.19 at pH = 12. However,
the size of the pearls is bigger and the elongation of the chain connecting them is
smaller compared to the same pearl-necklace observed for λ = 0.5. This is the result
56
of a weaker electrostatic interaction strength, that causes the repulsion effect between
equal charges to be less strong and for that reason the pearls are bigger. LJ = 2.5
titration curve has the same trend than for LJ = 1.5, except that it maintains a
constant decay until it reaches a zero value for pH = 14. This means that globule
states are present in a higher pH range than in previous LJ values. However we
might expect the presence of pearl-necklace like conformations at pH > 12, in the
region where charge asymmetry in the chain almost reaches its maximum value.
Figure 4.20: Radius of gyration for LJ = 0.5 at different λB values
Analysis of the radius of gyration was not performed for all LJ values for the three
different electrostatic strengths. The reason of this is the presence of pearl-necklacelike structures for LJ > 0.5, and to study those structures a different approach
technique has to be used. In order to analyze them, a separate analysis of the pearls
and the chain connecting them has to be carried out. Figure 4.20 shows the radius of
gyration of the whole chain for different λB values. As it can be noticed, the maximum
elongation of the chain occurs at the maximum charge asymmetry. Decreasing the
charge asymmetry decreases the size of the chain, and once we reach the isoelectric
point, the globule state is present. Figure 4.20 also illustrates the effect of weakening
57
the electrostatic interactions. Notice that the lower the value of λB , the lower the
maximum size the chain reaches. Finally, this figure also shows that the size reduction
of the chain presents a linear behavior, which might suggest that, the transition
between the globule and the extended chain is continuous.
4.4
Random polyampholyte in a high salt concentration
The random chain was also studied in the presence of a high salt concentration.
Same Debye screening length as in the diblock case was used corresponding to a salt
concentration cs ≈ 10−1 mol/L. Conformational diagrams were obtained for different
λB and LJ values.
Figure 4.21: Titration curves for type B monomers at different LJ and λB values in
the presence of a high salt concentration
Titration curves in Figure 4.21 show that in the presence of a high salt concentration, they have a similar behavior to the ideal case where no electrostatic interactions
are considered. They are also very similar to the titration curves for the diblock
58
case in the presence of salt illustrated in Figure 4.9, which as we mentioned in the
previous section, suggests that the distribution of the monomers along the chain does
not affect their dissociation behavior. Figure 4.22 shows the conformational diagram
for λB = 1. For LJ = 0.5, the chain presents globule-like conformations at pH values
close to the isoelectric point. As the charge asymmetry in the chain increases, the
chain extends and starts to behave like an ideal chain.
Figure 4.22: Conformational behavior of a weak random polyampholyte for λB = 1
and λD = 1
At the isoelectric point the charge asymmetry is equal to zero and the chain
exists in a globule conformation. In the presence of a high salt concentration, the
decay in the charge fraction of the monomers is faster than in the case where no
salt was present, meaning that the chain is very sensitive to pH changes. When
pH = 8, charge fraction of type B monomers was reduced by approximately 60% and
the globule-like structure start to break apart. As the pH keeps increasing, charge
asymmetry reaches a maximum value and the chain starts to behave like and ideal
59
chain. The conformation of the chain is described as a random walk state. Screening
effect of salt plays an important role in this behavior, because it does not allow the
chain to fully extend due to electrostatic repulsion between equal charges.
LJ = 1.5 and LJ = 2.5 titration curves show and identical trend as the LJ = 0.5
case with a smaller slope. However, because both chains are in a deeper hydrophobic
region and there is a screening effect in electrostatic interactions, the chain does not
present an extended state and it remains in a globule-like conformation for the whole
pH range.
and λD = 1
The conformational diagram of the chain for λB = 0.5 is present in Figure 4.23.
The LJ = 0.5 titration curve has the same behavior as for the λB = 1 case. Close to
the isoelectric point the chain stays in a globule state. In this case, the weakening of
the electrostatic interactions reduces the sensitivity of the chain to changes in the pH.
Meaning that globule-like conformations are stable for a larger pH range, as it can
60
be noticed for pH = 8, where the chain remains collapsed. As the charge asymmetry
increases, the polyampholyte shows signs of a random walk chain. However, because
the electrostatic interactions were weakened and salt screening effect is present, the
hydrophobic influence is more notorious and the chain shows a smaller size. Same as
in the previous case for λB = 1, LJ = 1.5 and LJ = 2.5 titration curves are almost
identical. The salt screening effect and the weaker electrostatic interactions cause
the dominance of the hydrophobic influence, keeping the polyampholyte in a globule
conformation even for high charge asymmetries.
and λD = 1
Finally, for λB = 0.25, the polyampholyte conformations are illustrated in Figure 4.24. Titration curves for the three LJ values have an identical behavior, which
means that the overall conformational state of the polyampholyte will depend on how
hydrophobic the chain is. LJ = 0.5 conformations show that the chain seems to be
in a collapsed state for the entire pH range. At the isoelectric point an interesting be-
61
havior of the chain is present, most of the chain seems to be in a globule conformation
with a small pearl necklace attached to it. By increasing the charge asymmetry, the
globule-like shape seems to remain stable for a wider pH range than in the previous
λB cases. Because the electrostatic interaction strength is very weak and there is salt
in the system, the hydrophobicity of the chain plays a more important role in the
shape of the chain. Even for a large charge asymmetry the chain seems to maintain
a collapsed-like shape instead of a random walk one. LJ = 1.5 and LJ shapes are all
globules because they are in a more hydrophobic region, and electrostatic interactions
are practically null.
Figure 4.25: Radius of gyration for LJ = 0.5 at different λB values in the presence
of a high salt concentration
The radius of gyration of the chain was only studied for the case LJ = 0.5 in
order to compare the results with the case where no salt was considered. Analysis of
the chain for LJ > 0.5 could have been done in this case because no pearl-necklaces
conformation are present. Figure 4.25 shows the radius of gyration of the chain for
LJ = 0.5 at different λB values in the presence of a high salt concentration. The
overall behavior of the chain is similar to the case where no salt was considered; at
high charge asymmetries the chain reaches its maximum size and at the isoelectric
62
point its minimum. Also, the maximum size of the chain decreases as the λB value
decreases. However, the only difference is that in the absence of salt, the size change
in the chain was more dependent on the charge asymmetry. In Figure 4.25 we can
see that the points are concentrated either in the maximum size or in the minimum
one. This might be the result of the screening effect in the electrostatic interactions
that reduces the pH range with a consequent high charge asymmetry.
63
Chapter 5
Concluding Remarks
5.1
Conclusions
A grand canonical MC molecular simulation has been implemented for a single polyampholyte chain. Two different configurations have been studied using a bead-spring
model and considering different electrostatic interaction strengths, hydrophobicities
and a high salt concentration. We did not model the solvent explicitly in our simulations. Instead, several considerations were done in order to account for the chemical
properties of the system. The study done here should be compared with studies were
counterion condensation plays an insignificant role.
The results have been used to demonstrate the conformational transitions that
a polyampholyte chain suffers depending on the solvent quality. In pure water, the
diblock polyampholyte shows globule to tadpole transitions depending on the hydrophobicity of the chain. In the presence of a high salt concentration, electrostatic
interactions are screened and the polyampholyte presents globule to random walk
transitions. For the random chain, globule, pearl-necklace and extended conformations are distributed along different pH and hydrophobic values. High ionic strength
screens the electrostatic interactions, where random walk and globule conformations
are the most stables, having a less chance to find pearl-necklaces in the whole pH
range. This research can be compared to experimental results on pH responsive
64
polyampholytes to determine the validity of our simulations.
5.2
Future Research Work
Recent studies have been performed to determine the complexes formed between
polyelectrolytes or nanoparticles and polyampholytes. Complexation between these
macromolecules seem to have an important role in DNA folding, medical imaging
techniques, enzyme entrapping and oil recovery processes [4, 16, 3, 9]. MC molecular simulations have been tested for complex formation between macromolecules and
results show that by varying the solution pH, the point where the complex formation is achieved can be tuned. The solvent quality for the polymer and electrostatic
interaction strength play an important role in determining the structure of a complex [16]. More research on this matter is needed in order to account for different
hydrophobic values for the polymer and to include the effect of salt concentration in
the complexation process.
65
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John Wiley & Sons, 2005,
69
APPENDICES
Appendix I: Derivation of the
Debye-Hückel equation
The Poisson’s differential equation can be used to relate the spatial distribution of
the electrostatic potential ψ(x) and the spatial distribution of charge density Qv (x)
[34]:
∇2 ψ = −
Qv (x)
ε0 ε
(A.1)
where ∇2 is the Laplace operator, ε0 is the permittivity of vacuum, and ε is the
dielectric constant of the medium considered. For the potential distribution around a
point charge, it is more convenient to use spherical coordinates, because all parameters
depend only on the distance r from the central ion [34]. In spherical coordinates, the
Poisson’s equation takes the form:
1 d(r2 (dψ/dr))
Qv
=−
2
r
dr
ε0 ε
(A.2)
A second equation is needed in order to link the parameters ψ and Qv . That
equation can be derived by assuming that the concentration of ions cj is determined
70
by the Boltzmann distribution law [34]:
cj =
z Fψ 0
j
0 − RT
cj e
(A.3)
where c0j is the concentration averaged over the entire volume and zj F ψ0 is the potential energy at a given point. The volume charge density depends on the ion distribution:
Qv (r) = F
X
zj cj (r) = F
X
zj cv,j e
−
zj F ψ0 (r)
RT
(A.4)
where the summation extends over all ions. Combining equations (A.1) and (A.4)
results in a second-order nonlinear differential equation for ψ0 (r). In the DebyeHückel theory, a simplified equation is used: Exponential terms in equation (A.4)
are expended into series and we only keep the first two terms of each series [34]. By
including the condition of electroneutrality and using the ionic strength Ic , equation
(A.4) can be rewritten as:
Qv (r) = F
X
zj c0j
2F 2 Ic
zj F ψ0
ψ0
=−
1−
RT
RT
(A.5)
By combining equations (A.1) and (A.5), we obtain the basic differential equation
of the Debye-Hückel theory:
1 d(r2 (dψ0 /dr))
= κ2 ψ0
2
r
dr
(A.6)
where parameter κ is defined as:
1
κ=
=F
λD
Equation (A.6) has the general solution:
2Ic
RT ε0 ε
1/2
(A.7)
71
ψ0 (r) =
C1 κr C2 −κr
e +
e
r
r
(A.8)
In order to find the constants C1 and C2 , we must formulate the boundary conditions. We know that at large distances from the central ion, the value of ψ is zero,
hence C1 = 0. To get the second coefficient, we substitute the second term in equation
(A.8) into equation (A.5) and integrate by parts from r = 0 to r = ∞ [34].
C2 =
zm Q0
4πε0 ε
(A.9)
Thus, equation (A.8) can be expressed as:
ψ0 (r) =
zm Q0 −κr
e
4πε0 ε
(A.10)
Equation (A.10) is the main equation in the Debye-Hückel theory, and it is used
to screen the electrostatic interactions, in the presence of salt, in our simulations.

ArturoMartinezJimene

Transcription

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