**5. Intrinsic viscosity of macromolecular coils and the thermal blob mass**

A macromolecular coil in a solution is modeled as an aggregate with mixed statistics consisting of *I* thermal blobs of 2 *DB* , each containing *Bi* solid monomers of radius *a* and mass *Ma* . To calculate the intrinsic viscosity

$$\mathbb{E}\left[\eta\right] \equiv \lim\_{\boldsymbol{c}\to\boldsymbol{0}} \frac{\eta - \eta\_{0}}{\eta\_{0}\boldsymbol{c}} \tag{22}$$

one has to define the mass concentration *c* of a macromolecular solution analyzed. The mass concentration in the coil, represented by the equivalent impermeable sphere, can be calculated as the product of the total number of non-porous monomers *<sup>B</sup> Ii* multiplied by their mass <sup>3</sup> 4/3 *<sup>s</sup> a* and divided by the hydrodynamic volume of the coil <sup>3</sup> 4/3*r* . This concentration multiplied by the volume fraction of equivalent aggregates gives the overall polymer mass concentration in the solution.

$$
\sigma = q \rho\_s \frac{I i\_B a^3}{r^3} \tag{23}
$$

Mass-radius relations are then employed. The thermal blob mass related to that of nonporous monomer is the aggregation number of the thermal blob

$$i\_B = \frac{M\_B}{M\_a} = \left(\frac{r\_B}{a}\right)^2\tag{24}$$

whereas the macromolecular mass related to that of thermal blob is the aggregation number of aggregate equivalent to coil

$$I = \frac{M}{M\_B} = \left(\frac{r}{r\_B}\right)^D \tag{25}$$

Hydrodynamic Properties of Aggregates with Complex Structure 259

2 2 2

*<sup>a</sup> c cc sa B <sup>B</sup> <sup>s</sup>*

The obtained equation can be also derived in terms of complex structure aggregate

<sup>5</sup> 3/ 1 3/ 1

*B s*

0

Equation derived for polymer coil can be compared to the empirical Mark-Houwink-

*MHS <sup>a</sup>*

5 5 2 2

*MM M*

1/2

*sa B sa MM M*

**1.e+4 1.e+5 1.e+6 1.e+7 1.e+8**

[ 

*M*a*MHS*

*M*

 *i I* 

2

5 2

*s*

Sakurada expression relating the intrinsic viscosity to the polymer molecular mass

 *K M*

*B*

 *K M*

*M*<sup>B</sup>

Fig. 5. Graphical representation of the Mark-Houwink-Sakurada expressions.

*a B*

*DB D*

3 1/ 3

*D*

*r r a r*

0

1/2 1/2 1/2

 

*c c M M M M M M*

 

*MHS*

 

(34)

1/2

*M M*

(35)

(36)

(38)

*<sup>M</sup>*

(37)

*B*

*MHS*

(33)

*a*

0

 

which is equivalent to

and

1/2 0 00 <sup>0</sup>

parameters for any blob fractal dimension to get

For the theta condition the formulae (Eq. 33) read

**1**

**10**

**100**

[

**1000**

**10000**

The Mark-Houwink-Sakurada expressions are presented in Fig. 5.

5 5 <sup>5</sup> lim lim lim

Taking into account that the volume fraction of polymer in an aggregate equivalent to polymer coil can be rearranged as follows

$$\frac{I\dot{i}\_B a^3}{r^3} = I \left(\frac{r\_B}{r}\right)^3 i\_B \left(\frac{a}{r\_B}\right)^3 \tag{26}$$

finally one gets

$$\mathbf{c} = \mathfrak{op} \rho\_s \left(\frac{M\_B}{M\_a}\right)^{-1/2} \left(\frac{M}{M\_B}\right)^{1-3/D} \tag{27}$$

or

$$\mathbf{c} = \eta \rho\_s \left(\frac{M\_B}{M\_a}\right)^{-1/2} \left(\frac{M}{M\_B}\right)^{-a\_{\rm MHS}} \tag{28}$$

if the fractal dimension *D* is replaced by the Mark-Houwink-Sakurada exponent *MHS a* , characterizing the thermodynamic quality of the solvent, where

$$a\_{\rm MHS} = \Im \left/ D - \mathbf{1} \right. \tag{29}$$

The structure of a dissolved macromolecule depends on the interaction with solvent and other macromolecules. The resultant interaction determines whether the monomers effectively attract or repel one another. Chains in a solvent at low temperatures are in collapsed conformation due to dominance of attractive interactions between monomers (poor solvent). At high temperatures, chains swell due to dominance of repulsive interactions (good solvent). At a special intermediate temperature (the theta temperature) chains are in ideal conformations because the attractive and repulsive interactions are equal. The exponent *MHS a* changes from 1/2 for theta solvents to 4/5 for good solvents, which corresponds to the fractal dimension range of from 2 to 5/3.

The viscosity of a dispersion containing impermeable spheres present at volume fraction can be described by the Einstein equation (Einstein, 1956)

$$
\eta = \eta\_0 \left( 1 + \frac{5}{2} \rho \right) \tag{30}
$$

from which

$$\frac{\eta - \eta\_0}{\eta\_0} = \frac{5}{2}\rho$$

The intrinsic viscosity can be thus calculated as

$$\mathbb{E}\left[\eta\right] = \lim\_{c \to 0} \frac{\eta - \eta\_0}{\eta\_0 c} = \lim\_{c \to 0} \frac{5}{2} \frac{\rho}{c} \tag{32}$$

Utilizing the expression for the mass concentration, one gets

$$\mathbb{E}\left[\eta\right] \equiv \lim\_{\mathcal{c}\to 0} \frac{\eta - \eta\_0}{\eta\_0 \mathcal{c}} = \lim\_{\mathcal{c}\to 0} \frac{5}{2} \frac{\mathcal{\phi}}{\mathcal{c}} = \lim\_{\mathcal{c}\to 0} \frac{5}{2} \frac{\mathcal{\phi}}{\mathcal{c}\rho\_s \left(\frac{M\_B}{M\_a}\right)^{-1/2} \left(\frac{M}{M\_B}\right)^{-a\_{\text{sites}}}} = \frac{5}{2} \mathcal{\phi}\_s \left(\frac{M\_B}{M\_a}\right)^{1/2} \left(\frac{M}{M\_B}\right)^{a\_{\text{sites}}}\tag{33}$$

The obtained equation can be also derived in terms of complex structure aggregate parameters for any blob fractal dimension to get

$$\left[\eta\right] = \frac{5}{2\,\rho\_s} \mathbf{i}\_{\rm B}^{\,3/D\_{\rm B}-1} \mathbf{I}^{3/D-1} \tag{34}$$

which is equivalent to

$$\mathbb{E}\left[\eta\right] = \frac{5}{2\rho\_s} \left(\frac{r\_0}{a}\right)^{3-1/D} \left(\frac{r}{r\_0}\right)^3 \tag{35}$$

Equation derived for polymer coil can be compared to the empirical Mark-Houwink-Sakurada expression relating the intrinsic viscosity to the polymer molecular mass

$$\mathbb{E}\left[\eta\right] = K\_{\eta}M^{a\_{\text{MHS}}} \tag{36}$$

For the theta condition the formulae (Eq. 33) read

$$\left[\eta\right]\_{\theta} = \frac{5}{2\,\rho\_s} \left(\frac{M\_B}{M\_a}\right)^{1/2} \left(\frac{M}{M\_B}\right)^{1/2} = \frac{5}{2\,\rho\_s} \left(\frac{M}{M\_a}\right)^{1/2} \tag{37}$$

and

258 Hydrodynamics – Advanced Topics

Taking into account that the volume fraction of polymer in an aggregate equivalent to

3 3 3

*a B M M*

*a B M M*

*M M*

if the fractal dimension *D* is replaced by the Mark-Houwink-Sakurada exponent *MHS a* ,

The structure of a dissolved macromolecule depends on the interaction with solvent and other macromolecules. The resultant interaction determines whether the monomers effectively attract or repel one another. Chains in a solvent at low temperatures are in collapsed conformation due to dominance of attractive interactions between monomers (poor solvent). At high temperatures, chains swell due to dominance of repulsive interactions (good solvent). At a special intermediate temperature (the theta temperature) chains are in ideal conformations because the attractive and repulsive interactions are equal. The exponent *MHS a* changes from 1/2 for theta solvents to 4/5 for good solvents, which

The viscosity of a dispersion containing impermeable spheres present at volume fraction

 

5 2

0 0 <sup>0</sup> <sup>5</sup> lim lim *c c c c* 2

 

0 <sup>5</sup> <sup>1</sup> 2

0 0

 

> 

<sup>0</sup>

Utilizing the expression for the mass concentration, one gets

 

*M M*

*Ii a r a I i r r r* 

*B B*

1/2 1 3/*D*

1/2 *MHS a*

(26)

(27)

(28)

*a D MHS* 3/ 1 (29)

(30)

(31)

(32)

3 *B B*

*<sup>B</sup> <sup>s</sup>*

*<sup>B</sup> <sup>s</sup>*

*c*

*c*

characterizing the thermodynamic quality of the solvent, where

corresponds to the fractal dimension range of from 2 to 5/3.

can be described by the Einstein equation (Einstein, 1956)

The intrinsic viscosity can be thus calculated as

polymer coil can be rearranged as follows

finally one gets

or

from which

$$\mathbb{E}\left[\eta\right]\_{\theta} = \mathbb{K}\_{\theta} M^{1/2} \tag{38}$$

The Mark-Houwink-Sakurada expressions are presented in Fig. 5.

Fig. 5. Graphical representation of the Mark-Houwink-Sakurada expressions.

Hydrodynamic Properties of Aggregates with Complex Structure 261

Let us imagine a coil consisting of one thermal blob. This is in fact a thermal blob of the structure of a large coil. Such rearranged blobs can join to another one to produce an object of double mass. The model makes it possible to calculate the fractal dimension *D* of the coil after each act of aggregation of two smaller identical coils of fractal dimension *Di* changing

Using the model for *D*lim 2 (the fractal dimension of thermal blobs), the dependences *i D <sup>B</sup>* have been calculated using CCA simulation, starting from both good and poor solvent regions. The aggregates growing by consecutive CCA events restructured to get a limiting fractal dimension *D*lim in an advanced stage of the process. Starting from 8 *Bi* and 5/3 *Di* , the result is *D*=1.8115. The second input to the model equation is thus 16 *Bi* and 1.8115 *Di* . Finally, the calculation results are presented in Fig. 6, where they

**1.6 1.7 1.8 1.9 2.0 2.1 2.2**

*D*

Fig. 6. Comparison of the model fractal dimension dependence of the thermal blob aggregation number (solid lines) to the representation of the experimental data measured

As discussed earlier, the ratio of the internal permeability and the square of aggregate radius is expected to be constant for aggregates of the same fractal dimension. Consider an early stage of aggregate growth in which the constancy of the normalized permeability is attained. At the beginning the aggregate consists of two and then several monomers. The number of pores and their size are of the order of aggregation number and monomer size, respectively. At a certain aggregation number, however, the size of new pores formed starts to be much larger than that formerly created. This means that the hydrodynamic structure building has been finished and the smaller pores become not active in the flow and can be

A part of the aggregate interior is effectively excluded from the fluid flow, so one can consider this part as the place of existence of impermeable objects greater than the monomers. Since both the impermeable object size and the pore size are greater than formerly, the real permeability is bigger than that calculated by a formula valid for a

for different polymer-solvent systems (Eq. 41), depicted as dashed lines.

**6. Hydrodynamic structure of fractal aggregates** 

regarded as connected to the interior of hydrodynamic blobs.

are compared to the dependence deduced from the empirical data.

**1**

**10**

**100**

*i*

*B*

**1000**

**10000**

with the aggregation progress.

There is a lower limit of the Mark-Houwink-Sakurada expression applicability. Intrinsic viscosity of a given polymer in a solvent crosses over to the theta result at a molecular mass which is the thermal blob molecular mass. This means that

$$\left(K\_{\eta}M\_{\text{B}}{}^{a\_{\text{MHS}}} = K\_{\theta}M\_{\text{B}}{}^{1/2}\right) \tag{39}$$

from which

$$M\_B = \left(\frac{K\_\theta}{K\_\eta}\right)^{1/\left(a\_{\rm{MHS}} - 1/2\right)}\tag{40}$$

The thermal blob mass depends on the Mark-Houwink-Sakurada constant at the theta temperature, characteristic for a given polymer-solvent system, as well as the constant and the Mark-Houwink-Sakurada exponent valid at a given temperature. The form of this dependence is strongly influenced by the mass of non-porous monomer *Ma* of thermal blobs, which is different for different polymers. The thermal blob mass normalized by the mass of non-porous monomer *MB a* / *M* , however, is the number of non-porous monomers in one thermal blob and therefore it expected to be a unique function of the solvent quality. This function, determined (Gmachowski, 2009a) from many experimental data measured for different polymer-solvent systems, reads

$$\dot{m}\_B = \frac{M\_B}{M\_a} = \left\{ \exp\left[0.9 \cdot \left(2a\_{\rm MHS} - 1\right)^{1/3}\right] \right\}^{a\_{\rm MHS}/\left(a\_{\rm MHS} - 0.5\right)}\tag{41}$$

The thermal blob aggregation number can be also calculated from the theoretical model of internal aggregation based o the cluster-cluster aggregation act equation (Gmachowski, 2009b)

$$i + i \sim \left[\frac{r}{R}(D)\right]^D \left(\mathbf{i}^{1/D\_i} + \mathbf{i}^{1/D\_i}\right)^D \tag{42}$$

being an extension of the mass-radius relation for single aggregate

$$\dot{a} = \left(\frac{r}{a}\right)^D = \left[\frac{r}{R}(D)\right]^D \left(\frac{R}{a}\right)^D \tag{43}$$

assuming it is a result of joining to two identical sub-clusters and its radius R is proportional to the sum of hydrodynamic radii 1/ 1/ *D D i i ai i* , where the normalized hydrodynamic radius is described by Eq. (5). Aggregation act equation can be specified to the form of an equality

$$\dot{\mathbf{i}}\_B + \dot{\mathbf{i}}\_B = \mathbf{2}^{1-D} \left[ \frac{r}{R} (D) / \frac{r}{R} (D\_{\text{lim}}) \right]^D \left( \dot{\mathbf{i}}\_B^{1/D\_i} + \dot{\mathbf{i}}\_B^{1/D\_i} \right)^D \tag{44}$$

for which *D* tends to *D*lim if *Bi* tends to infinity.

There is a lower limit of the Mark-Houwink-Sakurada expression applicability. Intrinsic viscosity of a given polymer in a solvent crosses over to the theta result at a molecular mass

*aMHS* 1/2 *KM KM*

The thermal blob mass depends on the Mark-Houwink-Sakurada constant at the theta temperature, characteristic for a given polymer-solvent system, as well as the constant and the Mark-Houwink-Sakurada exponent valid at a given temperature. The form of this dependence is strongly influenced by the mass of non-porous monomer *Ma* of thermal blobs, which is different for different polymers. The thermal blob mass normalized by the mass of non-porous monomer *MB a* / *M* , however, is the number of non-porous monomers in one thermal blob and therefore it expected to be a unique function of the solvent quality. This function, determined (Gmachowski, 2009a) from many experimental data measured for

/ 0.5 1/3 exp 0.9 2 1 *MHS MHS a a <sup>B</sup>*

The thermal blob aggregation number can be also calculated from the theoretical model of internal aggregation based o the cluster-cluster aggregation act equation (Gmachowski,

> 1/ 1/ ~ *i i <sup>D</sup> <sup>D</sup> <sup>r</sup> D D ii D i i*

 *D DD rr R i D aR a*

 <sup>1</sup> 1/ 1/ 2 / lim *i i <sup>D</sup> <sup>D</sup> <sup>D</sup> D D*

*B B B B r r ii D D i i R R* 

assuming it is a result of joining to two identical sub-clusters and its radius R is proportional to the sum of hydrodynamic radii 1/ 1/ *D D i i ai i* , where the normalized hydrodynamic radius is described by Eq. (5). Aggregation act equation can be specified to the form of an

   

1/ 1/2 *aMHS*

*B B* (39)

(42)

(43)

(44)

(41)

(40)

*B <sup>K</sup> <sup>M</sup> K* 

*B MHS*

*R* 

*<sup>M</sup> i a*

being an extension of the mass-radius relation for single aggregate

*a*

*M*

for which *D* tends to *D*lim if *Bi* tends to infinity.

which is the thermal blob molecular mass. This means that

different polymer-solvent systems, reads

from which

2009b)

equality

Let us imagine a coil consisting of one thermal blob. This is in fact a thermal blob of the structure of a large coil. Such rearranged blobs can join to another one to produce an object of double mass. The model makes it possible to calculate the fractal dimension *D* of the coil after each act of aggregation of two smaller identical coils of fractal dimension *Di* changing with the aggregation progress.

Using the model for *D*lim 2 (the fractal dimension of thermal blobs), the dependences *i D <sup>B</sup>* have been calculated using CCA simulation, starting from both good and poor solvent regions. The aggregates growing by consecutive CCA events restructured to get a limiting fractal dimension *D*lim in an advanced stage of the process. Starting from 8 *Bi* and 5/3 *Di* , the result is *D*=1.8115. The second input to the model equation is thus 16 *Bi* and 1.8115 *Di* . Finally, the calculation results are presented in Fig. 6, where they are compared to the dependence deduced from the empirical data.

Fig. 6. Comparison of the model fractal dimension dependence of the thermal blob aggregation number (solid lines) to the representation of the experimental data measured for different polymer-solvent systems (Eq. 41), depicted as dashed lines.
