Introduction

Comb-branched polymers with dense side chains are called brush-like polymers.1, 2, 3 After the establishment of synthetic methods, the number of studies on such polymers increased because of their interesting properties, such as liquid crystallinity.4, 5, 6 A typical method to obtain brush-like polymers is polymerization of macromonomers,7, 8 which consist of linear polymers with a polymerizable group at one end of the chain. From studies of polymacromonomers in solution, it was shown that these molecules behave as stiff chains.1, 2, 9, 10, 11, 12, 13, 14

We have studied the dimensional properties of polymacromonomers consisting of polystyrene (PS) in a good solvent (toluene at 15.0 °C) and in a theta solvent (cyclohexane at 34.5 °C) and determined the stiffness parameter λ−1 as functions of the degree of polymerization of side chain, ns.11, 12, 13, 15, 16, 17, 18 The λ−1 values obtained for these polymers were explained by the first-order perturbation calculation in terms of segmental interactions among side chains.19 It was also shown that the polymacromonomer chains expand because of intramolecular excluded-volume effects in a good solvent,12, 13 when the main chain is sufficiently long compared with the Kuhn segment length or λ−1. The magnitude of these effects are quantified by the excluded-volume parameter B,20, 21 which represents the excluded volume of a pair of segments divided by the square of the segment length. The B values were obtained from the radius expansion factor and found to increase with the side-chain length. The interaction between polymacromonomer segments may be described as a sum of segmental interactions among side chains.

In the calculation of λ−1 for brush-like polymers, the polymer molecule was modeled by a wormlike main chain with Gaussian side chains.19 Essentially the same model, a straight main chain with Gaussian side chains, may be used for the brush-like segment, which is an interacting unit of brush-like polymers. Interactions between two brush-like segments may be described by a sum of the interactions among side-chain beads. Because two side chains that belong to different brush-like segments contact at a place apart from the main chains, interactions between these segments are long-ranged. Thus, the problem is similar to segment–segment interactions of polyelectrolytes.22

The second virial coefficient A2 is also a reflection of the interaction between segments. Here, previously obtained data of A2 for PS polymacromonomers in toluene11, 15, 18 are analyzed to determine B values for different side-chain lengths. The values obtained for B are compared with the theoretical results calculated by the perturbation and the smoothed-density (SD) methods.

Theories

Basic equations

The brush-like segment is modeled by a straight main chain of length a with Gaussian side chains connected to it at intervals of hs. Each side chain consists of ns beads connected by bonds of length b.

The excluded volume for a pair of brush-like segments may be given by22

The inter-segment potential W12(r) depends on the directions of the main chains of the segments as well as the distance r between the middle points of the main chains. The angular brackets of the above equation signify averaging over the relative direction of these segments. The excluded-volume parameter B is defined by

The distance and directions of two brush-like segments are illustrated in Figure 1. The first brush-like segment (segment 1) is assumed to lie along the z axis of the Cartesian coordinates with its center of mass at the origin O. The side chains are numbered from 1 to a/2hs from the one closest to the origin for positive z and from −1 to −a/2hs for negative z. The side-chain beads (or segments) are numbered from 1 to ns from the closest one to the main chain. Here, the number of side chains in one segment a/hs and ns are considered to be much larger than 1. The position of the center of the main chain of the second brush-like segment (segment 2) is represented by r. The direction of segment 2 may be represented based on another set of Cartesian coordinates (x′,y′,z′) with its origin fixed at the center of segment 2 and with each axis being parallel to the first set of Cartesian coordinates. The azimuthal and rotational angles around the z′ axis are represented by θ and φ, respectively. The numbering of the side chains and the side-chain beads of segment 2 is same as that of segment 1.

Figure 1
figure 1

Coordinates of two brush-like segments.

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The probability that the pith bead of the ith side chain of segment 1 lies between R and R+dR may be given by the following Gaussian function:

Here, ri represents the position of the connecting point of the ith side chain to the main chain. A similar expression for P(R;rj,qj) for the qjth bead of the jth side chain of segment 2 can be obtained, where rj represents the distance from the origin O to the connecting point of the jth side chain to the main chain of segment 2.

The probability density that two segments pi and qj are in contact can be calculated as19

Here, rij=rj−ri and means that the distance between the segments pi and qj is zero. The vectors ri and rj−r can be written as follows:

Thus, rij is a function of r, i, j, θ, and φ, although it is not explicitely written.

Binary-cluster approximation

Perturbation calculation

The potential for two brush-like segments W12(r) may be expressed by the sum of the pair potentials between segments pi and qj, separated by .

Introducing the following function,

equation (1) can be expanded as23

Assuming short range interactions between segments, may be replaced by with the delta function, where β2 is the binary-cluster integral between side-chain beads. Then, the first term of the right-hand side of equation (7), designated by v2−1, can be written as

Substituting equation (4) into the above equation, v2−1 is obtained as

This is an obvious result because (nsa/hs)2 represents the total number of cases that one bead of segment 1 and another bead of segment 2 are in contact as shown in 2–1 of Figure 2.

Figure 2
figure 2

Diagrammatic representation of interactions among side chains. Thick and thin solid lines indicate the main and side chains, respectively. Dashed lines connect interacting segments.

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The second term of the right-hand side of equation (7) was also calculated. However, it was found that the absolute value of the second term is more than 10 times larger than the first term, indicating that the convergence of the power series is not good.

Smoothed-density model

Because the convergence of the perturbation calculation was not satisfactory, v is calculated according to the SD model. In this model, the inter-segment potential is written as23

Substituting the above equation with equation (4) into equation (1), the excluded-volume v2SD for this model is given by

The sums over i and j may be approximated by integrals from −∞ to ∞. Further, the following assumption is made according to Fixman and Skolnick.22 In Figure 1, the plane that is parallel to the z axis and contains the main chain of segment 2 is denoted as S. Assuming that the interaction between two brush-like segments occurs only when the projection of the main chain of segment 2 onto the xy plane crosses the perpendicular from the origin O to the plane S, the following is obtained:

Here, u represents the distance between the origin and the plane S. After summing over pi and qj, the excluded-volume v2SD is obtained by numerical integration over u and θ.

Effects of ternary clustering

Perturbation calculation

It is known that ternary-cluster terms are not negligibly small at and near the theta point as the binary-cluster integral reduces to the same order of magnitude as the ternary-cluster integral.24

Here, two ternary-clustering terms are considered. The first is for the case that two beads of a side chain belonging to segment 1 and one bead belonging to segment 2 contact, as shown in 3–1 of Figure 2, where the side chain belonging to segment 1 makes a loop. The term for this case may be written as

Here, the side chains i and j belong to segments 1 and 2, respectively. The segments pi and qj are on the side chain i and the segment sj is on the side chain j. The factor of 2 comes from the two cases that the side chain making a loop belongs to segment 1 or 2. The probability density in equation (13) is obtained with the aid of the Wang–Uhlenbeck theory23, 25 as

The sums in equation (13) over pi, qi and sj may be replaced by integrals from 0 to ns and those over i and j may be replaced by integrals from −∞ to ∞. After performing these integrations, along with those on r, θ and φ, v3−1 is obtained as

Here, σ means the minimum number of segments for making a loop. If v3−1 is combined with v2−1, the following is obtained:

The expression in the brackets of the above equation is the same as the one that appears in the second virial coefficient of linear polymers.24, 26 It is known that the negative β2 and the positive β3 terms cancel out at the theta point, where A2 becomes zero, by24, 26

Thus, it is expected that the contribution from v2−1+v3−1 to B vanishes at this point for sufficiently large ns.

Next, the following case is considered: two beads belonging to different side chains of segment 1 and one bead belonging to segment 2 contact, as shown in 3–2 of Figure 2. The term for this case may be written as

Here, the side chains i and j belong to segment 1 and the side chain k to segment 2. The factor of 2 appears for the same reason as in equation (13). The probability density for this case can be written as

Substituting equation (18) into equation (17), v3−2 is obtained as

Here, all the summations were approximated by integrals.

Smoothed-density model

The contribution of the ternary-clustering to v may also be calculated by the SD model. If the interactions of three beads belonging to different side chains are considered, the potential for segments 1 and 2 may be given by

From a similar argument to that of the SD model in the binary-cluster approximation, the following expression of v3 for this model is obtained:

The factor of 2 in front of β3 in the above equation appears for the same reason as it does in equation (17).

Comparison with experiments

Good solvent systems

Previous A2 data for PS polymacromonomers with different side-chain lengths in toluene at 15.0 °C12, 15, 18 are plotted against the weight-average molecular weight Mw in Figure 3. Here, F15, F33, F65 and F110 represent polymacromonomers consisting of side chains with 15, 33, 65 and 113 styrene residues, respectively. A2 values smaller than 10−5 cm3 mol g−2 were not included in the figure because they are less accurate. It is seen that A2 decreases with Mw and that the slope becomes steeper with increasing side-chain length.

Figure 3
figure 3

Molecular weight dependence of the second virial coefficient for polystyrene polymacromonomers with different degrees of polymerization of the side chain in toluene: circles, F15;12 filled triangles, F33;12 unfilled triangles, F65;15 squares, F110;18 lines, calculated values.

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These data may be explained by the following equation for the wormlike chain,27

Here, A02 represents the term without the contribution of the chain-end effect and may be calculated from the following equation for the wormlike chain21

where NA, L and M denote the Avogadro constant, the contour length of the polymacromonomer molecule and molecular weight, respectively. The last two parameters can be related by L=M/ML with the molecular weight per unit contour length ML. The dimensionless variable h(λB,λL) may be calculated from the modified Barrett equation.21, 27, 28

The term A2(E) for the chain-end effects may be expressed as27

Here, a2,1 and a2,2 are constants. The second term of the right-hand side of equation (24) is neglected here.

B and a2,1 are determined by fitting calculated curves to the data points as illustrated by the solid lines in Figure 3, where known values of λ−1 and ML for each sample12, 15, 18 were used. The variable hs was taken to be 0.26 nm, the averaged value for these samples.12, 15, 18 The value β2=0.034 nm3 known for linear PS in toluene at 15.0 °C29 and b=0.74 nm calculated from the unperturbed dimension of PS in cyclohexane at 34.5 °C30 were also used. The value for the highest molecular weight sample of F65 is less accurate because the measurements were taken at low concentrations, therefore, it was omitted from the fitting. The resulting values are summarized in Table 1.

Table 1 Molecular parameters for polystyrene polymacromonomers in toluene at 15.0 °C
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The circles in Figure 4 show the values of B thus determined plotted against ns. The values obtained from the current analyses are close to those determined from 〈S2〉 shown by the squares. The dashed line in Figure 4 represents the calculated values from equation (9). It is seen that they are much larger than the experimental values. The calculated values from the SD model represented by the solid line are much closer to the experimental data. This result suggests that the segment distribution around the main chain of polymacromonomer molecules can be described by the Gaussian function even if the conformation of each side chain with small ns is far from Gaussian.

Figure 4
figure 4

B for polystyrene polymacromonomers determined from the radius expansion factor (squares)12 and the second virial coefficient (circles). Dashed and solid lines show calculated values from equation (2) with equations (9) and (12), respectively.

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Theta solvent systems

In Table 2, calculated B values from v2−1 and v2SD according to equations (19) and (21), respectively, are summarized. Here, β3=0.004 nm3 is used, as determined from the third virial coefficients of linear PS in cyclohexane at 34.5 °C.24 It is seen that the values from the perturbation calculation are much larger than those for the SD model. If we ignore the chain-end effect, the maximum values of A2 may be calculated from equation (23) with h(λB,λL)=1. The A2 values obtained with B based on the SD model were less than 2 × 10−5 cm3 mol g−2 except for F15. From the previous results, the absolute values of A2 for PS polymacromonomers in cyclohexane were less than 3 × 10−5 cm3 mol g−2 at the theta temperature (34.5 °C) irrespective of molecular weight.11, 15, 18 Thus, the calculated values for F33, F65 and F110 by the SD model agree with the experimental results. The value for F15 was about 5 × 10−5 cm3 mol g−2, which is rather larger than the experimental values.11 This suggests that it is necessary to consider some other interactions, such as the residual β3 interaction corresponding to the ns−1/2 term in the square brackets of equation (16), although it cannot be taken into consideration by the SD model. The first-order perturbation calculations gave much larger A2 values (order of 10−4 cm3 mol g−2) than those from the SD model, showing the importance of the closed equation form.

Table 2 Calculated B for polystyrene polymacromonomers in cyclohexane at 34.5 °C
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Conclusion

In this study, theoretical investigations of B for brush-like polymers in good and theta solvents were carried out. It was shown that the SD-model calculations gave closer values to the experimental values in both good and theta solvent systems if binary and ternary interactions among side-chain beads were taken into consideration. The first-order perturbation calculation of B gave much larger values than the observed ones. This result is similar to the zeroth-order approximation of A2 for linear flexible polymers.23 The calculated value for the β22 term of B was much larger than the β2 term, showing that the convergence of the power series in terms of β2 is not good. The β22 terms for B arise from the double interactions between side-chain beads belonging to different brush-like segments. Such higher-order interactions are considered to occur frequently when brush-like segments approach each other in an almost parallel configuration. For the same reason, the SD-model calculation with β3 interactions gave much closer values to the experimental ones for theta solvent systems than the first-order perturbation calculation. However, those values from the SD model calculations still deviate from experimental values by 10–50%. To obtain quantitative agreements, Monte Carlo calculations of B for the brush-like segments may be desirable.