Polymers in solution phases have a high degree of freedom for translational and inter-nal motion. They change their conformations randomly by Brownian movements. The purpose of this section is to see how these molecular characteristics of polymers lead to the macroscopic properties of the polymer solutions.
Measurements to find the characteristics of each polymer chain are carried out by separating themfromeach other in solution. Knowing the fundamental properties of polymer solutions, in particular dilute solutions, is very important for the molecular
characterization of polymers. Studies on dilute polymer solutions have historically played an important role in polymer science [6]. In addition, concentrated polymer solutions have practical applications to polymer processing. This section provides an overview of the nature of polymer solutions with a comparison to solutions of low-molecular weight solutes.
The following conventional units of concentration will be used:
• Mole fraction xi= ni/
ini
The number of i-component molecules/total number of molecules in the mixture (dimensionless).
• Volume fraction φi= Vi/V
The volume occupied by i-component molecules/total volume of the mixture (dimensionless).
• Weight % wi= niMi/
iniMi
The weight of thei-component/total weight of the mixture (dimensionless [wt %]).
• Molarity mi= ni/n0M0
The number ofi-component molecules in a unit weight of the solvent ([mol kg−1]).
• Mole concentration νi= ni/V
The number of moles of thei-component in a unit volume of the mixture ([mol dm−3]).
• Weight concentration ci= niMi/V
The mass of thei-component in a unit volume of the mixture ([kg dm−3]).
For simplicity, let us assume that the volumea3of a statistical repeat unit of a polymer is the same as that of a solute molecule. If there areN0solvent molecules andN1polymer chains of the lengthn in terms of the number of repeat units in a volume V =(N0+nN1)a3 of the solution, the concentration is
ν =N1
V , c =mnN1
V , x = N1
N0+N1
, φ = nN1
N0+nN1
,
in terms of the unit given above, wherem is the mass of a repeat unit.
The characteristic feature of polymer solutions is that they largely deviate from the ideal solution. We will look at this step-by-step in the following subsections.
2.2.1 Vapor pressure and osmotic pressure
The vapor pressure of a polymer solution deviates downwards largely from the Raoult’s law. Figure2.6plots the pressure of the solvent vapor in equilibriumwith a solution of uncrosslinked rubber in benzene. The depression becomes larger with the molecular weight of the polymer. Thus, polymers of high molecular weights significantly suppress the solvent activity.
As for the osmotic pressure of a polymer solution, the first term c/M in the virial expansion (2.39) is small due to the factorM−1when compared with the same concen-trationc of a low-molecular weight counterpart. The second virial coefficient A2, which
Mole Fraction x
Activity a0=p0/p0o
Ideal solution 1
0.5
0
0 0.5 1
M = 103
M = 3 × 105
Fig. 2.6 Vapor pressure depression of polymer solutions. The solvent activity (vapor pressure normalized by the reference value of the pure solvent) is plotted against the molar fraction of the polymers.
50°C
40°C
30°C 1.8
1.6
1.4
1.2
1.0
0 0.5 1.0 1.5 2.0
π/c
c[g/100 cm3]
Fig. 2.7 Osmotic pressure of polystyrene/cyclohexane solutions. The molecular weight of polystyrene is 203 000. The figures beside the curves show their temperatures (Reprinted with permission from Krigbaum, W. R., J. Am. Chem. Soc. 76, 3758 (1954).)
originates in the interchain interaction, is very important. Figure 2.7plotsπ/c of the solutions of polystyrene in cyclohexane against the weight concentrationc. In the limit of dilutionc → 0, we can find RT /M, and hence we can find the molecular weight M of the polymer. The initial slopes of these lines giveA2. Its sign changes between the temperatures 30◦C and 40◦C fromnegative to positive. The temperature at which the condition
A2(T ) = 0 (2.49)
Table 2.1 Theta temperature of common polymer solutions
Polymer Solvent Theta temperature [◦C]
polyethylene diphenylether 161.4
polystyrene decalin 31
cyclohexane 34.5
polypropylene cyclohexanone 92
isoamyl acetate 34
poly(vinyl chloride) benzyl alcohol 155.4
Poly(methyl methacrylate) 2-heptanone 11
acetonytril 30
2-octanone 52
poly(dimethyl siloxane) methylethylketon 20
chlorobenzene 68
is fulfilled is called the theta temperature1 of a solution, and denoted T = 8. The theta temperature is fixed by the combination of polymer and solvent. It takes a dif-ferent value for the same polymer if the solvent is difdif-ferent. The theta temperature of polystyrene/cyclohexane is8=34.5◦C. Table2.1summarizes the theta temperatures of common polymer solutions.
We can roughly estimate the second virial coefficient of the osmotic pressure by regarding a polymer as a rigid sphere with the same radiusR as the radius of gyration of the randomcoil (thermodynamic equivalent sphere). As shown in Figure2.8,A2of the hard sphere systemis the volume of the region where the sphere cannot enter due to the presence of other spheres. It is equal to the volume of the spherical region with the radius 2R, the diameter of the rigid sphere, and hence we have
A2=4π
3 (2R)3 1
M2, (2.50)
whereM2in the denominator is required to change the number of the sphere to the mass densityc used in the definition. Because the radius of gyration of a randomcoil in a good solvent isR ∼ Mν, we have
A2∼ M3ν−2, (2.51)
and hence the power lawA2∼M−0.2holds for the swollen chain with the Flory exponent ν = 3/5. Experiments report that the exponent lies in the range 0.1–0.5, with a typical value of 0.2. Detailed calculation of the second virial coefficient on the basis of the perturbation expansion is presented in the classic textbook by Yamakawa [7].
1 In this book, we discriminate it from the molecular theta temperatureθ defined in Chapter1based on the intramolecular interaction.8 depends on both intra- and intermolecular interaction. If the interaction between the statistical repeat units can be described by a single excluded volume parameterv in (1.71), these two are identical. In the perturbational calculation of the third virial coefficient, simple substitution of (1.71) cannot explain the observation of positiveA3> 0 at the 8 temperature. In such a case, the third cluster integral must be introduced in addition to the binary cluster integralv.
2R
Fig. 2.8 Excluded volume (broken line) between the equivalent spheres representing polymer chains. The second virial coefficient of the osmotic pressure is proportional to the excluded volume.
2.2.2 Viscosity
The viscosity of a liquid is defined as follows. Keep a liquid between the two parallel plates, and apply a forceσxyper unit area to the upper plate in thex-direction perpendic-ular to they-axis. The force σxyis called the shear stress. The first index indicates the force direction, and the second indicates the direction of the normal vector perpendicular to the surface. The liquid flows in thex-direction, and the stationary velocity field vx(y) with a constant velocity gradient (shear rate)
˙γ ≡∂vx
∂y , (2.52)
is established after a sufficiently long time (Figure2.9).
The stationary viscosityη is defined by the ratio of the shear stress to the velocity gradient
η( ˙γ) =σxy
˙γ . (2.53)
The viscosityη of the polymer solution depends in general on the shear rate ˙γ. The term
“viscosity” usually indicatesη( ˙γ) in the limit of the small shear rate ˙γ → 0, η0≡ lim
˙γ→0η( ˙γ). (2.54)
Whenever its dependence on the shear rate is studied,η( ˙γ) is referred to as the nonlinear stationary viscosity. The CGS unit of the viscosity [g cms−1] is called poise. Its MKS unit is [kg ms−1] ≡ [Pa s].
The viscosity is related to the energy dissipation in the liquid. Letd be the separation between the two plates, and let us consider the lower part with areaS. The upper area moves by a distance ˙γd in the x-direction per unit time, and hence the stress does the work( ˙γd)(σxyS) on the liquid. This work is dissipated as heat generated by the friction
Area S
(a) (b)
y
z x
d
γd γd
σxy
υx
Fig. 2.9 (a) Shear flow and viscosity of the solution. Polymers flow toward downstream while they rotate.
(b) The work done by the shear stress in a unit time for the solution to flow.
between the molecules in the liquid. By definition, the stress isσxy= η ˙γ, and the work done by the stress is( ˙γd)(σxyS) = (η ˙γ2)(Sd), so that the heat quantity generated in a unit time in a unit volume is proportional to the viscosity asη ˙γ2.
The small shear rate region where the nonlinear viscosity is independent of the shear rate is called Newtonian region. With an increase in the shear rate, the viscosity of ordinary polymer solutions decreases. This phenomenon is known as shear thinning.
In polymer solutions in which polymers associate with each other by strongly attractive forces, such as hydrogen bonding, hydrophobic association, etc., the viscosity increases with the shear rate, reaches a maximum, and then decreases. The increase of the viscosity by shear is called shear thickening. Typical examples of thickening solutions are solutions of associating polymers. Shear thickening caused by nonlinear stretching of the polymer chains will be studied in Chapter9.
The viscosityη of a solution is a function of the concentration. Its increment due to the solvent relative to the reference valueη0of the pure solvent is the specific viscosity
ηsp≡η −η0
η0
. (2.55)
The specific viscosity is proportional to the concentration in the dilute region; the reduced viscosity defined byηred≡ ηsp/c is often used. It can be developed in a power series of the concentration:
ηsp
c = [η]+k2c +k3c2+··· . (2.56) The first term[η] is the intrinsic viscosity (or limiting viscosity number). It has the dimension of the reciprocal of concentration, and has a value of order unity when measured by the unit of g dm−3. More precisely,
[η]c∗ 1, (2.57)
Dimensionless Shear Rate τγ.
Relative Viscosity
[ η
]/[η]0
(1) (2) (3) (4)
(7)
(6)
(5) 1.00
0.90
0.80
0.70
0.60
0.1 1 10
Fig. 2.10 Relative intrinsic viscosity as a function of the shear rate: poly(α-methylstyrene) in toluene with molecular weight is (1) 690 k, (2) 1240 k, (3) 1460 k (4) 1820 k, (5) 7500 k, polystyrene with a molecular of weight 13 000 k in toluene, (6) and in decalin (7) The viscosity exhibits shear thinning phenomena. The Newtonian plateau region depends on the molecular weight. (Reprinted with permission from Noda, I.; Yamada, Y.; Nagasawa, M., J. Phys. Chem. 72, 2890 (1968).)
wherec∗is the overlap concentration, the concentration at which polymer random coils start to overlap with each other. The overlap concentration will be described in detail in Section2.4.1.
Figure2.10shows an example of the viscosity of a polymer solution measured as a function of the shear rate. The relative intrinsic viscosity[η]( ˙γ)/[η]( ˙γ = 0) is plotted against the reduced shear rateτ ˙γ, where τ is the characteristic relaxation time. Crossover fromthe Newtonian region to the thinning region can be seen.
The coefficient of the second termk2gives the effect of hydrodynamic interaction between two polymer chains. The interaction is mediated by the flow of the solvent around them. The strength of the hydrodynamic interaction is usually described by the dimensionless number called the Huggins coefficient:
kH≡ k2/[η]2. (2.58)
In commonly occurring polymer solutions, the Huggins coefficient takes the value in the range 0.3–0.7 (see Figure2.11).
The intrinsic viscosity contains the information on the conformation and molecular motion of each individual polymer chain. It depends on the molecular weight in the power law (the Mark–Houwink–Sakurada relation)
[η] = KMa, (2.59)
where the Sakurada constant K is a constant depending on the combination of the polymer and solvent. The power indexa takes a value in the range 0.5–0.8. Table2.2 lists several examples.
ηsp/c
concentration c[g dl–1] 3.1
2.9
2.7
2.5
2.30 0.04 0.08 0.12 0.16 0.20 0.24 (ln (η/η0))/c
Fig. 2.11 Specific viscosity of polystyrene in benzene plotted against the polymer concentration [g dm−3].
The molecular weight of the polymer isMw= 360000.
Table 2.2 Intrinsic viscosity and the molecular weight
Temp. [◦C] K 103
Polymer Solvent [dm3g−1] a
polystyrene cyclohexane 34.5 84.6 0.50
butanone 25 39 0.58
(cis-)polybutadiene benzene 30 33.7 0.72
poly(ethyl acrylate) acetone 25 51 0.59
poly(methyl methacrylate) acetone 20 55 0.73
poly(vinyl acetate) benzene 30 22 0.65
poly(tetrahydrofuran) toluene 28 25.1 0.78
Let us derive the relation (2.59) by comparing the random coil of a polymer with a hard sphere. It is known for a suspension of rigid hard spheres of massm and volume v that the specific viscosity is given by
ηsp=5
2φ +κ2φ2+··· , (2.60)
whereφ ≡Nv/V is the volume fraction of the spheres in the suspension. The coefficient 5/2 was found by Einstein in 1906. The exact value of the second coefficient κ2is difficult to find, but is estimated to be 7.6 from the approximate solution of the hydrodynamic equation. Becauseφ = vc/m, we find that [η] = 5v/2m by comparing this equation with (2.56). The intrinsic viscosity depends on the mass density m/v of the sphere and is independent of the total mass (molecular weight). Hence we havea = 0.
Let us assume the random coil in the solution as a hard sphere of the radiusRHas in the thermodynamic sphere (Figure2.8). This hypothetical sphere is not the representative of the segment distribution, but shows the region inside the coil where the solvent flow cannot pervade. It is called the hydrodynamically equivalent sphere (Figure2.12). Its volume isvH= 4πR3H/3. The radius RHis not the same as the radius of gyration, but is
Fig. 2.12 Hydrodynamically equivalent sphere defined by the region into which the solvent flow does not pervade.
expected to be proportional to it. Therefore, let us introduce the proportionality constant by the relationRH= λ s2 1/2. The constantλ shows the degree of solvent pervasion. A uniformrigid sphere hasλ = (5/2)1/2= 1.58, while a randomGaussian coil has a value of the order ofλ 0.69. Fromthe Einstein coefficient for the rigid sphere, we find
[η] = 2.5vH
m = 0 s2 3/2
M . (2.61)
Because the relationsm = M/NA,vH= 4πRH3/3 hold, we find 0 = 2.5×4πλ3NA. This constant is known to take the value0 = (2.1±0.2)×1023g−1mol−1by measurement, and is regarded as a universal constant independent of the materials studied.
Because (2.61) can be transformed to [η] =
0( s2 0
M )3/2
M3ν−1, (2.62)
the Sakurada constant is given by
K = 0( s2 0/M)3/2, (2.63)
and the indexa is a = 3ν −1. (The subscript 0 indicates a Gaussian coil. The radius of gyration s2 0of a Gaussian chain is proportional to the molecular weightM.) K is a constant independent ofM. At the theta temperature, polymer chains can be regarded as Gaussian withν =0.5, and hence a =0.5. At high temperatures where chains are swollen by the excluded volume effect with the Flory index, ν = 3/5, and hence a = 0.8. The experimental results summarized in Table2.2can thus be explained.
2.2.3 Diffusion of a polymer chain
If the concentration is not uniformin the solution, but depends on the position, the solute molecules diffuse from regions of high concentration to regions of low concentration.
This is due to random Brownian motion of the solute molecules.
×
Fig. 2.13 Molecular diffusion. (a) Counting the number of molecules moving across a hypothetical unit area in the solution. (b) Counting the number of molecules entering and exiting the region between parallel planes separated by an infinitesimal distancedx.
The mass of the solute molecules moving across an infinitesimal areadS in the solution in a unit time is given by J·ndS, where n is the unit normal vector perpendicular to this area, and J is the flux vector. For instance, the mass flux of solute molecules moving across the unit area perpendicular to thex-axis is Jx.
The flux is proportional to the gradient of the concentration,
J= −D∇c, (2.64)
because of the diffusion. This is Fick’s law. The negative sign shows that the diffusion takes place froma region of high concentration to a region of low concentration. The proportionality constantD is the diffusion constant. It is a material constant of the solute molecules in a given solvent.
To derive Fick’s law, consider a fictitious plane perpendicular to the x-axis at the positionx, and count the number of molecules that cross the small area dS on this plane (Figure2.13(a)). To describe the random Brownian motion of the solute molecules due to thermal agitation, let us assume for simplicity that each molecule moves by one step of widtha in a fixed short time τ in randomdirections with equal probability. Because there are three axes in the space, and each axis has± direction, on average 1/6 of the total molecules move to the+ direction of the x-axis. The number of molecules that pass the area during the time intervalτ is therefore 1/6 of the molecules in the cylindorical volumeadS in the left-hand side of the plane. If the number density in the volume is represented byn(x −a/2,t) at the central position P(x −a/2) of the volume, then a total
of 1
6n(x −a/2,t)adS
molecules cross the area to the positive direction. A similar formula holds for the molecules moving in the negative direction. Taking the difference and dividing by the area, we find
for the number flux to the positive direction. The multiplication of the mass of a molecule to this equation leads to Fick’s law for the mass flux Jx≡ mjx. Hence the diffusion constant is given by
D =a2
6τ. (2.66)
(The squared step lengtha2divided by the fundamental time scaleτ necessary for one step of movement.) The number 6 comes from the space dimensionsd multiplied by 2 for the± directions. The diffusion of a marked particle obtained in such a way is the self-diffusion constant or marker diffusion constant.
Let us next count the number of molecules that are entering and exiting the region between the parallel planes at the positionx and x + dx separated by an infinitesimal distancedx in the system(Figure2.13(b)). Because massJx(x,t) enters fromthe left-hand plane per unit area per unit time, and massJx(x +dx,t) exits fromthe right-hand plane, the mass inside the region changes by
∂
∂t(cdx) = Jx(x,t)−Jx(x +dx,t) −∂Jx
∂xdx.
Substituting Fick’s law (2.64) into this equation, we find that the concentration obeys the diffusion equation
∂c
∂t = D∂2c
∂x2. (2.67)
If we observe the displacement of a Brownian particle over a long time intervalt, it looks like the conformation of a random flight polymer chain with a fundamental step length a and number of repeat units n = t/τ (Figure1.4). The displacement R of the particle corresponds to the end-to-end distance, and its square average should be equal to
R2 = na2=ta2
τ = 6Dt. (2.68)
(For thex-component, the relation is x2 = 2Dt.)
The diffusion constant is related to the friction of the particle with the media. Letζ be the friction constant of the diffusing particle. Einstein found that the relation
D =kBT
ζ (2.69)
holds, wherekBis the Boltzmann constant (Einstein relation).
When a rigid sphere of radius a moves in a solvent of viscosity η0, the friction coefficient is given by Stokes’law
ζ = 6πaη0. (2.70)
In the case of polymers diffusing in a solvent, we can replace the random coil by the hydrodynamically equivalent sphere of radiusRH. We find
ζ = 6πη0RH, (2.71)
Table 2.3 Some examples of diffusion constants
Solute Molecular weight Solvent D [10−7cm2s−1]
sodiumchloride 58 water 80.0
polystyrene 10 600 benzene 11.7
polystyrene 67 000 benzene 4.1
polystyrene 606 000 benzene 1.5
and hence the friction coefficient should obey the power lawζ ∝ Mν. The molecular weight of the polymer can therefore be estimated by measuring the diffusion constant.
The absolute values of the diffusion constant for different materials are in the range 10−7–10−6cm2s−1. Table2.3shows some examples.
The diffusion constant obtained by tracing the selected particle among many is the marker diffusion constant. The marker diffusion constant is indicated by the labeling symbol *, asD∗. In contrast, the diffusion constant in Fick’s law is defined for the many particles involved in the local concentration, and is called the concentration diffusion coefficient. In dilute solutions where particles move independently of each other, these two diffusion constants are the same. In concentrated solutions, the assumption of inde-pendent motion of the particles breaks down by molecular interaction, so that the two diffusion coefficients are not identical.
To study the concentration diffusion coefficient, let us focus on a solute particle in solution. Its average velocity u is decided by the balance condition between the thermal driving force−∇µ and the viscous resistance force ζ ¯u,
ζ ¯u = −∇µ, (2.72)
whereµ(r,t) is the chemical potential of the particle at the position r, and ζ is the friction constant. The mass flux J= c ¯u then takes the form
J= −(c/ζ)∇µ = −(c/ζ )(∂µ/∂c)T∇c, (2.73) and hence the concentration diffusion coefficient is given by
D =
c ζ
∂µ
∂c
T
. (2.74)
Although the marker diffusion coefficient is always positive, the concentration diffu-sion coefficient may become negative when the thermodynamic instability condition (∂µ/∂c)T < 0 is fulfilled (Section2.3). Particles spontaneously move from the regions of low concentration to the regions of high concentration. When a solution is quenched froma high-temperature uniformstate to a low-temperature unstable state in the spinodal region, it separates into two phases through such negative diffusion. The method used to observe the time development of the phase separation process by such a temperature quenching is called the spinodal decomposition method.
The chemical potential of the solute particles in a solution can be written as
µ(c,T ) = µ0(T )+kBT ln(γ c), (2.75) (whereµ0(T ) is the reference value) by using the activity coefficient γ .The concentration
µ(c,T ) = µ0(T )+kBT ln(γ c), (2.75) (whereµ0(T ) is the reference value) by using the activity coefficient γ .The concentration