Now,finally time to talk about Carbon nanotube dispersion. Here I explained the basic thermodynamics of dispersion for carbon nanotubes. Critical thing to point out is, carbon nanotube aspect ratio gives raise to entropy of mixing term. And the energy spent to separate macromolecules from each other is much larger than spherical particles. For this reason, they are evaluated to rigid-rod structures for their entropy calculations.
I believe this song will be okay while reading 🙂 right ? 16sya1
Apart from below, there is one study from Bergin.at al, who was claiming about entalpy of a solvent for carbon nanotube mixture regarding to be negative(for NMP, 2-methyl-pyrolidone), but apparently the formulations proove that it can not be negative. I just want to warn people who investigate this area.If you search for carbon nanotube thermodynamics, most probably will come across that paper and feel thousands of question marks as I experienced. Do not believe anything you read! Question and try to understand why it is so. Equations tell us that entalphy of mixture can not be negative. Hope below research will help you to understand carbon nanotubes.
Thermodynamic Basics and Solubility Parameters for Carbon Nanotubes
Fazlı Fatih Melemez
Sabanci University Engineering Faculty and Natural Sciences
Abstract: In this study, dispersion state of carbon nanotubes are investigated thermodynamically under the light of different interaction parameters. Hildebrand and Hansen solubility parameters are generally used parameters to evaluate dispersion of polymers in solvents. It is concluded that neither hansen or hildebrand parameters are fundamental to predict and evaluate dispersion state for carbon nanotubes and for CNT dispersion prediction, established numerical tool yet to be established. Keywords:Carbon nanotube, dispersion state, dispersion prediction, Hansen Solubility Parameter.
Contents
Abstract . ………………………………………………………………………………………………………………………….. 1
Contents ………………………………………………………………………………………………………………………………… 1
1.Introduction ……………………………………………………………………………………………………………………. 2
2.Dispersion Enhancement Treatments : Surface Modifications ………………………………………………. 2
2.1.Mechanical Surface Modifications ( Ball Milling ) ……………………………………………………….. 2
2.2.Physicochemical Surface Modifications ……………………………………………………………………… 3
2.2.1.Covalent Surface Modifications ………………………………………………………………………….. 3
2.2.2.Non-Covalent Surface Modifications …………………………………………………………………… 3
2.2.3.Irradiation ………………………………………………………………………………………………………… 3
2.3.Evaluation of the Dispersion of CNTs ………………………………………………………………………… 4
3.Thermodynamic Approach ……………………………………………………………………………………………….. 5
3.1.Flory-Huggins Interaction Parameter ………………………………………………………………………….. 5
3.2.General Thermodynamics …………………………………………………………………………………………. 8
3.3.Entropic and Entalpic Treatment of Carbon Nanotubes ……………………………………………….. 10
4.Hansen Solubility Parameters @ Hansen Space ………………………………………………………………… 13
APPENDIX-A ……………………………………………………………………………………………………………………… 15
References …………………………………………………………………………………………………………………………… 16
_______
1. Introduction
It has been more than twenty years after one of the most surprizing discovery of the material science environment. Ijima , when he was first discovered the carbon nanotubes using transmission electron microscopy, who contributed a lot to get a stride from theoretical studies to experimental observations. After his remarkable contribution numerous studies are performed in the field to characterize carbon nanotubes with respect to their chemical,mechanical, electronical and optoelectronical properties. CNTs, regardless of its various types, show promising applications which may increase the current state-of-art rapidly at each of this diverse but inter-connected fields. Our research is majored on to reveal state-of-the-art for solubility properties of CNTs, which is one of the most important and strongest hurdles on CNT applications.
2. Dispersion Enhancement Treatments : Surface Modifications
2.1. Mechanical Surface Modifications ( Ball Milling )
A typical process including ball milling CNTs that are degassed by heating under a nitrogen atmosphere or under vacuum with purging by a reactant gas for the duration of ball milling process. As in the most surface modification process, ball milling also reduces the tube length of CNT. It lowers the aspect ratio of CNTs and percolation to be hardly formed, which is the unwanted situation because it causes interruption for stress transfer in nanocomposite or composites mechanical enchancement applications and conductivity problem for electrical applications.
Figure1.SEM images and length distributions of ball-milled MWCNTDNA conjugates. (a)Pristine MWCNT, (b) MWCNT–
DNA conjugates, (c) length distribution, and (d) photographs of solutions containing MWCNT DNA conjugates: 5mg/mL, 0.25 mg/mL, pristine MWCNT in water, and DNA (from left to right) [1].
2.2. Physicochemical Surface Modifications
2.2.1. Covalent Surface Modifications
This type of modification is preferred for capacitors or catalyst supports to increase compability of the interface with metal oxides and to improve the chemical stability of the composite materials[2].
These are the well-known types of covalent surface modification techniques: Oxidation, Halogenation, Cycloaddition, Radical Addition, Electrophilic Addition, Thiolation, Electro-chemical reduction, Nucleophilic cyclopropanation, Amidation & Esterification.
2.2.2. Non-Covalent Surface Modifications
In contrast with covalent surface modifications which locally disrupt sp2 hybridization or defect creation and CNT destruction on the walls, non-covalent surface modifications are advantegous on preserving sp2-conjugated hybridized, electronic structure of CNTs. Adsorption or wrapping are the mechanisms of non-covalent modifications.
2.2.3. Irradiation
Gama and plasma radiation have some advantages when they are used in conjunction with conventional wet chemical routes. These advantages can be counted as time saving, less pollution and easy control over reaction degree at lower temperatures[1]. Plasma surface modification is particularly interesting due to its flexibility,contaminant-free and relatively non-desctructive nature[3],[4],[5].
Apart from those methods, electron beam radiation is also used to modify CNT surfaces for their effective dispersion in polymer matrices. However EB irradiation induces some structural changes onto CNTs such as colescence,welding,exfoliation,cross-linking, amorphization and cutting [6].
2.3. Evaluation of the Dispersion of CNTs
-potential measurements determine the electrical potential at the surface of CNTs and can be used to monitor the dispersion state and stability of CNTs because CNT colloids are dispersed and stabilized by electrostatic repulsion force between the surface charges on neighbouring nanotubes.
Any variation on the surface charge induced electrostatic repulsion furce can be translated to stability measure. zeta–potential indicates the degree of repulsion between adjacent,similarly charged particles. For molecules and particles that are small enough, a high zeta –potential will confer stability so that solution or dispersion will resist agglomeration. When the potential is low, attraction exceeds repulsion and the dispersion will break and flocculate. zeta -potential is determined from the electrophoretic mobility by using the Henry‟s Equation:
where \nu is the solution viscosity, \mu is the electrophoretic mobility of the charged particle, is the dielectric constant of the solvent, \’k’ is the debye-huckel parameter and a is the radius of the charged particle.
Since it is difficult to determine “ka” for CNTs, they are classified as rod like particles and Smoluchowski‟s approxiation is usually adopted to estimate -potential of CNTs with following equation;
Figure4. Potential difference as af function of distance from particle surface (left),Zeta potential values and their physical meaning for a solution(left) [8].
In general, colloidal particles with -potential >15 mV are expected to be stable. It may be a bit of representative parameter for CNT/surfactant suspensions but even though the -potential is obtained from Smoluchowski‟s approximation, It may 20 % overestimate the value when it is applied to rod-like particles such as CNTs [9]. Additionally, there is currently no absolute quantitative tools for measuring dispersion quality.
Because of reasons counted above, its not possible to make well-established generalization to a rigid relation between zeta –potential and solubility of CNTs. It is used to describe surface characteristics but not to describe solution stability in a surrounding media. Flory-huggins interaction parameter may provide a better descirption of the affinity between CNTs or the miscibility with surrounding media[1].
3. Thermodynamic Approach
3.1. Flory-Huggins Interaction Parameter
Ideal behaviour of mixing is idenfitied by Raoult‟s Law in the sense that both solute and solvent molecules are that of comparable size. But in case of macromolecules (long chain molecules such as CNTs) Flory-Huggins solution theory plays role. It is a mathematical model of the thermodynamics of polymer solutions which takes account of the great dissimilarity in molecular sizes in adapting the usual expression for the entropy of mixing.
Flory-Huggins polymer solution theory is developed after 1941 by the referred scientistists independently to develop a simple lattice model theory. Their model was leaning to assumptions as following;
Low molecular weight solvent,
Low molecular weight solute,
Solvent and solute have same molecular size,
Let‟s consider a solution which is composed of two N1 and N2 components. And each small regions in the solution must be occupied by exactly one molecule of either solvent or solute component. So, we can say that solution is formed by periodic sites. So , total number of these sites are given as;
N1 is the number of solvent molecules and N2 is number of polymer (solute)molecules, each of which has x segments.
From statistical mechanics, we can calculate the entropy change;
Where k is Boltzmann‟s constant. And we define the lattice volume fractions;
O1 and O2 are also probabilities which defines existance of that phase in a given lattice site. For a small solute whose molecules occupy just one lattice site, x = 1.This is pretty much complex for carbon nanotubes whose molecular structure occupies more than one site.
Using the boltzman relation for increase in entropy due to mixing, using the probability function and sterling approximation will lead to the expression for the molar entropy change of mixing for an ideal solution;
Xi is mole fraction of solvent and R is the gas constant. Then, they considered another experiment with the following identities;
Low molecular weight solvent
High molecular weight polymer
And with these assumptions, they gave an expression for enthalpy of mixing;
In such a mixture, there are three types of interaction to be considered; \ double-v_11 represents molecular interaction between monomer-monomer,\double_v_22 represents the same for solvent-solvent and \double_v_12 for solvent-monomer interaction. Energy increment per monomer-solvent contact is;
And total number of monomer-solvent contacts in the system is given by;
where z is coordination number (number of nearest neighbour for a lattice site,each one occupied by one chain segment or a solvent molecule). xN2 is total number of monomers in the solution. Here, It should be noted that N1 is the matrix (solvent phase) and N2 is solute, which means for a high quality dispersion N2 molecules should be totally independent from each other and they homogenously should be distributed across the matrix, which also means that number of N2 molecules are equal to sub regions (segments) in the solution which is given by . Meaning that x = 1 for solution which has one solute in it.
In light of these informations, we can write the entalphy change which is equal to energy change per polymer-monomer solvent interaction multiplied by the number of such interactions;
Then they defined a single, dimensionless, energy parameter called Flory-Huggins Interaction parameter,X12 ,which is inversely proportional to temperature but independent on concentration
As it is seen from above equation, flory-huggins interaction parameter is depending on nature of both solvent and solute through mean field approximated molecular interactions. Mean field approximation is a tool of mean field theory, also called self-consistent field theory, has the main idea that to evaluate one particle in a multi body environment (at which every particle interact with each other), focuses only to one particle and assumes that most important contribution of interaction to that particle through nearest-neighbouring particles. These theory used in many field to overcome computational challenges such as Ising and Heisenberg models in magnetism, Landau theory in physics and so on.
If we insert value of ” 
” from the interaction parameter type into Enthalpy equation, it yields to;
This term equals to enthalpy change of the mixture,N1 N2 . If we convert molecules to moles, n1 and n2, write the total free energy change, it is like the following;
Here the value of the flory-huggins interaction parameter can be estimated either from Hildebrand or Hansen Solubility Parameters. From Hildebrand solubility parameters 
of materials;
where Vseg is the actual volume of a polymer segment. 
are material solubility parameters and they‟re given as square root of cohesive energy density; 
. This equation also tells us why in our daily life, probability for a liquid to be dissolved in another one is more favoured. Because in this case, cohesive energy densities of given liquids are much closer each other and interaction parameter is less, thus entalpy has lower value. Which also means amount of energy spent to seperate all molecules from their neighbours is relatively less than that of second phase is a solid.
3.2. General Thermodynamics
Gibbs Free Energy describes the potential of a system to do non-mechanical work, meaning that is a measure of „process-initiating work‟ capacity of a system at constant temperature and pressure.
Under the light of these information, we can conclude that, if the Gibbs free energy of a mixture is negative, solute phase is said to be “soluble”. Which means “dispersion”.
Gibbs free energy of mixing determines whether mixing at constant T and P, is a spontaneous process. For an ideal solution, entalpy of mixing is equal to zero since it is assumed that there is no interaction between molecules, so energy spent to take them apart from each other is zero. Therefore, for ideal case it is equivalent to entropy of mixing. But in the case CNTs solution, situation is exactly the opposite of it. CNT molecules are relatively big molecules due to their high aspect ratio, entropy of mixing value of their solutions is queitly large and energy to take them apart from each other is huge. And for most of the cases, entalpy is positive value. That‟s the explanation of why we have suspension most of the cases.
Entalpy is the total energy content of the system or component, includes the internal energy and the energy is added to the system by external work. Entalpy of mixture is summation of its components such that,
Entropy is used to determine available energy can be converted to useful work in a system. For a mixture, entropy of mixing term is concluded as below;
Now, what is entropy of mixture and entalpy of a mixture. If we dwell on these terms, we will better understand the characteristics of the solution. It is naturally enforced that entropy of mixing is always positive value. By definition, entalpy of mixing is, the measure of energy spent to seperate solute molecules from individual solvent molecules, this value can be positive or negative depending on the system.
Entalpy of mixture can be calculated if the system is composed of small molecules. It is calculated either with Flory-huggins parameter or Hildebrand-Stracthard Expression. These parameters are identified through the medium of two material‟s solubility parameters. So, to be able to calculate interaction parameter, one should know the solubility parameter of materials according to either Hildebrand or Hansen solubility approach (no other approaches settled yet).
If we recall this formulation, 
, flory-huggins parameter is positive for most of the materials, results in a positive value of entalpy of mixture. As a result, we can say by remembering the main equation, , most solutions are driven by the positive entropy term and for this reason, they form suspension. Basically, it means if change in entalpy of mixture is greater than term, solution is named as suspension. Otherwise, it is named as dispersion. Which is the state for gibbs free energy change to be negative.
3.3. Entropic and Entalpic Treatment of Carbon Nanotubes
Carbon nanotubes are definitely not small molecules. They have very high aspect ratio (length/diameter). So, their molecular size is quietly large. From the following formulations, we can see why carbon nanotubes or rod-like molecular structures have relatively small entropy.
[formulations are written on the basis of per volume of a mixture]
So this formulations tells us that, entropy of the nanotubes because of their large size and high aspect ratio is quietly small(absolute value is large but its sign makes it quietly small). For this reason, to make Gibbs free energy negative, entalpy of carbon nanotube and solvent solution should be minimum. It is now aim to understand at which conditions entalpy is minimized. When Flory-Huggins interaction parameter is small. Or alternatively, when 
.
Now, we are approaching to Hansen’s parameter explanation. As we mentioned before hildebrand parameter is based on dispersion and only applicable to molecules with conducted by London forces. But in most of the molecular interactions, polarity and hydrogen bonds also effect the solubility. Here, Hansen stepped forward with calculation of cohesive energy densities of dispersive, polar and hydrogene bonding components, such that;
This cohesive energy terms leads us to three solubility parameters that all of them are extracted from square root of the associated cohesive energy density terms. They are known as Hansen Solubility Parameters and shown as following;
Under the light of Hansen solubility parameters, Flory-Huggins parameter is evaluated and written again;
Above equation answers our question regarding to when entalpy is minimized. It means that entalpy of mixture is minimized when solute-and-solvent have similar values for all three hansen parameters. It is quietly same situation as in the case of Hildebrand approach of interaction parameter. At both, x minimizes the entalpy of mixture.
4. State of the Art for Carbon Nanotube Dispersion
Highest reported concentrations were known as below;
SWNT in Dimethyformamide (DMF) solvent, 0.125 mg/ml.
SWNT in Tetrachloroethylene solvent, 0.07 mg/ml,
SWNT in Dichlorobenzene, 0.095 mg/ml.
Here, wait a moment and look at the above datas, how small they are. And these are records (by the end of 2011) what science world holds in hand.
Without surface treatment, the results come out from the tests of nanotube dispersions so far are not so promising. However, in the very recently published review [12], authors mention about an optimized sonication conditions (Coleman and Hamilton, unpublished work) and following centrifuge technique which enables greatly dispersed and stable dispersions. It is explained as following; “This procedure is known to give high quality dispersions containing only individual and small bundles of nanotubes. For example we have observed CHP,NMP and NbenP to display less than 6 nm bundle size at reasonably high concentrations. ” Table 1 shows the best 14 solvent tested with this unique centrifugation.
According to Table.1., CHP has extraordinary dispersion with single walled carbon nanotubes as much as 3.5 mg/ml.
For sake of application, one can ask the following crucial question.
How can we predict about a solvent if it is a good dispersive agent for our needs ? A study [10],[13] shows that apart from the interaction parameter calculations, surface tension of solvent molecules is also related to dispersability such that they form a dispersion when the surface tension is closer to that of graphite, which is around 40 MJ/m2. Again, it is not perfect solubility parameter,for example NMP has perfect match with graphite yet it does show much lower solubility as seen from Table 1. Simplest general description of a good solvent is one whose Hildebrand parameter matches that of solute.
Figure5. Maximum dispersibility, Cmax, for both solvents and nonsolvents as a function of (A) solvent surface tension and (B) solvent Hildebrand parameter. Nonsolvents are those with dispersibilities of effectively zero. The vertical dashed lines illustrate the weighted mean of (A) the surface tension and (B) the Hildebrand parameter[12].
5. Hansen Solubility Parameters @ Hansen Space
However, only a fraction of solvents with the correct Hildebrand solubility parameter disperse nanotubes. Thus, it is necessary to describe a set of solubility parameters, namely, Hansen Solubility Parameters.
Figure 6. (A) Nanotube dispersibility as a function of the dispersive Hansen parameter, (B) polar Hansen parameter and (C) hydrogen bonding Hansen parameter.
Here one who wants to learn dispersability of a solute-solvent system should know the Hansen Solubility parameters of both solute and solvent. There are some ways explained in the literature for both carbon nanotube and solvents [14],[15].
Lets assume we have the solute and solvent‟s Hansen Solubility parameters. What is the final comparison criteria between these parameters. It is defined as distance in Hansen space from the point representing solvent solubility parameters and solute solubility parameter. Since we defined good solvent as having solubility parameter as close as possible to that of solute, R has the same meaning in Hansen Space. If they‟re closer , polymer is more likely dissolved in that solvent (~90%).
Three parameters (dispersion,polar and hydrogen) can be treated as co-ordinates for a point in three dimensions also known as the Hansen space. The nearer two molecules are in this three dimensional space, the more likely they are to dissolve into each other. To determine if the parameters of two molecules (usually a solvent and a polymer) are within range a value called interaction radius (R0) is given to the substance being dissolved. This value determines the radius of the sphere in Hansen space and its center is the three Hansen parameters. To calculate the distance (Ra) between Hansen parameters in Hansen space the following formula is used;
In order to explain why Hansen solubility parameters are needed, we should look what really Hansen and Hildebrand SP‟s does. Hildebrand solubility parameter as we said, dependent to directly cohesive energy density per volume. Cohesive energy density means the heat to vaporize that molecular structure (amount of heat to break all bonds forms that structure) for a given volume. But Hansen approach also takes the polar and hydrogene bonding term into account and calculates the cohesive energies for them as well. For example, nitroethane and ethanol have almost same cohesive energy density but their solubility is totally significantly differed [16].
APPENDIX-A
Gibbs Potential _ Gibbs Free Energy
It is a thermodynamic potential of a system to do non-mechanical work. For one component, it is electrochemical potential. For a mixture, electrochemical potential is additive over constituents and it is summation of them.
Gibbs Free Energy of Mixing
For a mixture, gibbs free energy of a system should be always negative. It is a measure of the system to initiate a process whether spontaneously or not.
Entalpy
Entalpy is total amount of energy belonging to one system.Covering internal energy and work capability.
Entalpy of Mixing
This is generally positive value in daily life. For ideal case, since any molecul or atom talks to each other, energy spent to take them apart is zero. So, for effective nano dispersion it should be negative or so small. But in CNT case, entalpy formulation for rigid-rod like structures, are depending on the volume segment of CNT(which is pretty big), that‟s why we mostly have positive entalpy with CNT mixtures. There is only NMP has negative value. But it has again very poor dispersion of CNT‟s.
Entropy
Useless energy content of a system. When heat is supplied on a system, it will be spent on work and entropy. If the system is fixed meaning that mechanical work is constrained, it will totally spent on entropy.
Entropy of Mixing
Since it is additive, it is calculated through additivity with an updated version which is shown above. For rigid rod structures, formulation is given above which has second and third terms boost the entropy of mixing value through high aspect-ratio of CNT‟s. Entropy is always positive value, eventhough it is shown as negative, it has ln function inside which corresponds to negative quantity since ln function is negative between 0 and 1.
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8. http://en.wikipedia.org/wiki/Zeta_potential
9. Wang P, Shi Q, Liang H, Steuerman DW, Stucky GD, Keller AA. Enhanced environmental mobility of carbon nanotubes in the presence of humic acid and their removal from aqueous solution. Small 2008;4(12):2166–70]. 10. Ausman, K. D.; Piner, R.; Lourie, O.; Ruoff, R. S.; Korobov, M. Organic Solvent Dispersions of Single-Walled Carbon Nanotubes: Toward Solutions of Pristine Nanotubes. J. Phys. Chem. B 2000, 104, 8911–8915 11. Multicomponent Solubility Parameters for Single-Walled Carbon Nanotube−Solvent Mixtures Shane D. Bergin, Zhenyu Sun, David Rickard, Philip V. Streich, James P. Hamilton, and Jonathan N. Coleman, ACS Nano 2009 3 (8), 2340-2350
12. Carpenter, L.; Blau, W. J.; Boland, J. J.; Hamilton, J. P.; Coleman, J. N. Towards Solutions of Single-Walled Carbon Nanotubes in Common Solvents. Adv. Mater. 2008, 20,1876–1881]
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