Kramer theorem: Branching Ratio Derivations

Instead of using Zimm-Stockmayer expressions that relate the branching ratio 𝑔 with the number of branch points of (only) ideal randomly branched structures, it is possible to derive exact relations for polymers with known shapes by using Kramers theorem. Because the polymer shapes in the used models are well defined, it is important to find accurate relations between branching ratio 𝑔 and the number of branch points in order to create accurate polymer ensembles. In this appendix, the derived 𝑔-factor expressions that are obtained from Kramers theorem are worked out. Unique expressions are acquired for star, comb and cayley tree shaped polymers.

Kramers Theorem

calculated. Alternatively, the Kramers theorem can be expressed in the following way:

< 𝑅_𝑔²>= ^𝑏2

𝑁𝑡𝑜𝑡𝑎𝑙< 𝑁(𝑁_{𝑡𝑜𝑡𝑎𝑙}− 𝑁) >

For a linear polymer, the radius of gyration will be:

< 𝑁(𝑁_{𝑡𝑜𝑡𝑎𝑙}− 𝑁) > =_𝑁¹∫ 𝑁(𝑁₀^𝑁 𝑡𝑜𝑡𝑎𝑙− 𝑁)𝑑𝑁

This result is the same for the classic Debye result for the radius of gyration of an ideal linear chain.

With this the generalized branching ratio 𝑔 can be defined for Kramer theorem, when using

Figure 2 Kramers theorem evaluates all different options to divide a polymer into two. 𝑵(𝒌) andn 𝑵𝒕𝒐𝒕𝒂𝒍− 𝑵(𝒌) represent the monomers that are present in the two polymer fractions.

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A1.1 Star Shaped Polymers

Star shaped polymers consist of 𝑥 identical arms that are all connected to one point and all have one free end. The summation in the generalized branching ratio 𝑔 (shown below) can be simplified into a single summation that considers one linear arm. This summation times the number of arms results in the total contribution of the arm segments to the branching ratio 𝑔 (Eq. A1.1.1).

𝑔 = _𝑁⁶

𝑡𝑜𝑡𝑎𝑙3 ∑^𝑁_𝑘=1𝑁(𝑘)[𝑁_{𝑡𝑜𝑡𝑎𝑙}− 𝑁(𝑘)]

𝑔 = _𝑁⁶₃∗ 𝑎𝑟𝑚𝑠 ∑^𝑛_𝑖=1𝑖(𝑁 − 𝑖) (A1.1.1)

Here 𝑁 is the total length (or total number of monomers), 𝑛 is the arm length and 𝑖 is the monomer coordinate. Because 𝑛 is large, the summation can be changed into an integral: ∑^𝑛_𝑖=1→∫ 𝑑𝑙_𝑜^𝑛 . Here 𝑙 is the continuous coordinate that moves along the arm segment.

𝑔 = ⁶

Because all polymer arms have the same length, 𝑛 can be defined in terms of 𝑁 and the number of arms (Eq. A1.1.3). Adding this to equation A1.1.2 gives 𝑔 in terms of the number of arms. This result resembles the derivations of Zimm-Stockmayer [20].

𝑛 = ^𝑁

The branching ratio 𝑔 is typically defined in terms of number of branch points 𝑏. For comb or cayley tree structures, two segments are added when the number of branch points is increased by one.

Furthermore the structure for comb and cayley when 𝑏 = 1 is a three-armed star. Therefore, for star shaped polymers, we start with a three-armed star and add 2 arms when the number of branch points is increased, i.e. 𝑎𝑟𝑚𝑠 = 2𝑏 + 1. Inserting this into equation A1.1.4 gives the final result for star shaped polymers.

𝑔 =_(2𝑏+1)^6𝑏+1₂ (A1.1.4)

47

A1.2 Comb Shaped Polymers

In this section branching ratio 𝑔 as a function of the number of branch points 𝑏 is derived from Kramers theorem for ideal comb shaped polymers with equal segment lengths. We start with the generalized 𝑔𝑘𝑟 function (Eq. A1.1).

𝑔 = _𝑁⁶

𝑡𝑜𝑡𝑎𝑙3 ∑^𝑁_𝑘=1𝑁(𝑘)[𝑁_{𝑡𝑜𝑡𝑎𝑙}− 𝑁(𝑘)]

The structure of comb polymers for increasing number of branch points 𝑏 is shown in figure 2. Here a couple of trends can be found. Firstly, the comb structure can be separated into two parts:

backbone segements and branch segments. Branch segments contain a free end and the number of branch segments is equal to the number of branch points. Backbone segments lie deeper in the polymer with increasing number of branch points. The connectivity value (or color value in figure 2) given to the segments is determined from finding the amount of segments on either side of segment 𝑥 and taking the smallest value of the two. For example the encircled segment that lies in the comb structure with 5 branch points has 4 segments on its left side and 6 segments on its right side, resulting in a connectivity value of 4. These connectivity values are evenly distributed for

symmetrical comb structures (with uneven number of branch points). Evaluating both symmetrical and unsymmetrical structures would lead to more elaborate functions (Eq. A1.2.10). Therefore we select variable 𝑎 that accounts for the number of symmetrical comb structures (Eq. A1.2.2).

𝑎 =^𝑏+1₂ (A1.2.2)

Because the comb structures can be split into two contributing parts, Eq. A1.2.1 is separated in a function for the backbone 𝛾_𝑏𝑏 and a function for the branches 𝛾_𝑏𝑟.

𝑔_{𝑘𝑟,𝑐𝑜𝑚𝑏}= ⁶

𝑁³(𝛾_𝑏𝑏+ 𝛾_𝑏𝑟) (A1.2.3)

When using Kramers theorem for the encircled segment 𝑥 in figure 2, the first option of dividing the polymer into two is by separating the left 4 segments from segment 𝑥 plus the right 6 segments. In general, segment 𝑥 can be split at position 𝑖 that lies between 0 and 𝑛. At 𝑖 = 𝑛 the polymer is split into the left 4 segments plus segment 𝑥 from the right 6 segments. The summation for this segment can be defined as∑^5𝑛_4𝑛𝑖(𝑁 − 𝑖) or ∑^𝑛_𝑖=1(𝑖 + 4𝑛)(𝑁 − (𝑖 + 4𝑛)), where 𝑛 is the segment length and 𝑁 the total length. The connectivity value scales with variable 𝑎 with 2(𝑗 − 1), where 𝑗 is the connectivity number (red = 1, blue = 2, green = 3 etc.).

Figure 3 Structure of Comb shaped polymer for different number of branches. 𝒃 is the number of branches and 𝒂 is the variable that considers only symmetric comb polymers. The legend in the top right corner indicates what connectivity value each polymer segment has. The solid lines represent the backbone and the dashed lines branches.

48 With this the summation for 𝑦𝑏𝑏 is obtained and shown in equation A1.2.4. The value 2 in front of the second summations accounts for symmetry of the comb, where every connectivity value occurs twice.

𝛾_𝑏𝑏 = ∑^𝑎_𝑗=12 ∑^𝑛_𝑖=1(𝑖 + 2𝑛(𝑗 − 1)) (𝑁 − (𝑖 + 2𝑛(𝑗 − 1))) (A1.2.4) Because the polymer segments are of equal length, the segments length is only dependent on the total length and the number of branch points. In equation A1.2.5 𝑛 is be defined in terms of 𝑁 and 𝑎.

𝑛 = ^𝑁

4𝑎−1 (A1.2.5)

The second summation in Eq. A1.2.5 is solved first and the result is shown in equations A1.2.6.

𝑦 = ∑^𝑛_𝑖=1(𝑖 + 2𝑛(𝑗 − 1)) (𝑁 − (𝑖 + 2𝑛(𝑗 − 1)))

𝑦 = ∑^𝑛_𝑖=1(𝑖𝑁 − 𝑖²− 4𝑖𝑛𝑗 + 4𝑖𝑛 + 2𝑛𝑁𝑗 − 4𝑛²𝑗²+ 8𝑛²𝑗 − 2𝑛𝑁 − 4𝑛²)

When 𝑛 is sufficiently large, the summation can be changed into an integral: ∑^𝑛_𝑖=1→∫ 𝑑𝑙_𝑜^𝑛 . Here monomer coordinate 𝑖 is replaced by continuous coordinate 𝑙 that moves along the segment chain.

𝑦 = ∫ (𝑙𝑁 − 𝑙₀^{𝑛 2}− 4𝑙𝑛𝑗 + 4𝑙𝑛 + 2𝑛𝑁𝑗 − 4𝑛²𝑗²+ 8𝑛²𝑗 − 2𝑛𝑁 − 4𝑛²)𝑑𝑙

Here it is not possible to change the summation into an integral because the variable 𝑎 is not large enough. Therefore the following summation expressions have to be used:

∑^𝑣_𝑢=11= 𝑣

49 For the branch segments the summation function 𝛾𝑏𝑟 is simple. Because the branch segments have a free end, the summation is taken from 0 to 𝑛. The number of branch points is equal to the number of branch points. The number of branch points in terms of a is 𝑏 = 2𝑎 − 1. Equation A1.2.8 describes 𝑦𝑏𝑟. Filling in 𝑛 and 𝑎 gives the final result.

𝑛 =_4𝑎−1^𝑁

If both even and uneven numbers of branches are included before the derivation of 𝑦𝑏𝑏 equation A1.2.10 could to be used. In equation A1.2.9 we assume that the even branched comb polymers have a value between its neighboring uneven values and fit the equation, so in the end the branching ratios of even branched comb polymers can be estimated with this equation too. Starting from A1.2.10 might be more ‘complete’, but it is expected that the difference in values for even combs between A1.2.9 and A1.2.10 is very minimal. This is why we start from equation A1.2.5.

𝛾_𝑏𝑏= (∑ 2 ∑^𝑛_𝑖=1(𝑖 + 2𝑛(𝑗 − 1)) (𝑁 − (𝑖 + 2𝑛(𝑗 − 1)))

𝑏+1−(−1)𝑏

𝑗=12 ) + (¹

2(1 + (−1)^𝑏) ∑^𝑛_𝑖=1(𝑖 + 𝑛(𝑏 + 1)) (𝑁 − (𝑖 + 𝑛(𝑏 + 1)))) (A1.2.10)

50

A1.3 Cayley Tree Structured Polymers

Here branching ratio 𝑔 as a function of the number of branch points 𝑏 is derived from Kramers theorem for ideal cayley tree shaped polymers with equal segment lengths. Similar with the previous part, we start with the generalized 𝑔 function derived from Kramers theorem (Eq. A1.1).

𝑔 = _𝑁⁶

𝑡𝑜𝑡𝑎𝑙3 ∑^𝑁_𝑘=1𝑁(𝑘)[𝑁_{𝑡𝑜𝑡𝑎𝑙}− 𝑁(𝑘)]

In model 1, cayley trees were built from branch points that were added in a clockwise manner. Here however symmetry is desired to prevent the formula from becoming highly complicated. In

literature cayley trees are typically defined in terms of generations. Figure 3 displays schematically the structure of cayley trees for increasing generations. For generation number 𝑔𝑒𝑛 = 1 a three-armed polymer is used. The next generation is obtained when adding two segments to the free ends of the previous generation. The connectivity value (the color value in figure 3) scales according to 2^𝑗− 2, where 𝑗 is the connectivity number (red = 1, blue = 2, green = 3 etc.). The amount of segment with connectivity number 𝑥 scales with the connectivity number and the generation number, 3 ∗ 2^{𝑔𝑒𝑛−𝑗}. With this the 𝑔 relations is obtained:

𝑔 = ⁶

𝑁³∑^𝑔𝑒𝑛_𝑗=13 ∗ 2^{𝑔𝑒𝑛−𝑗}∑^𝑛_𝑖=1(𝑖 + (2^𝑗− 2)𝑛) ∗ (𝑁 − (𝑖 + (2^𝑗− 2)𝑛)) (A1.3.1) 𝑁 is the total number of monomers, 𝑔𝑒𝑛 is the cayley tree generation number, 𝑗 is the connectivity number, 𝑖 is the segment monomer coordinate and 𝑛 is the number of segment monomers. Because the polymer segments are of equal length, the segments length is only dependent on the total number of monomers and the generation number. In equation A1.3.2 𝑛 is be defined in terms of 𝑁 and 𝑎.

𝑛 = ^𝑁

3(2^𝑔𝑒𝑛−1) (A1.3.2)

First the second summation of Eq. A1.3.1 is solved. Because 𝑛 is large the summation is changed into an integral, with 𝑙 being the continuous coordinate along the segment chain.

0

Figure 4 Structure of Cayley Tree shaped polymer for different generation numbers. 𝒃 is the number of branches and 𝒈𝒆𝒏 is the generation number. The legend in the top right corner indicates what connectivity value each polymer segment has.

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𝑔 =_𝑁⁶₃∑^𝑔𝑒𝑛_𝑗=13 ∗ 2^{𝑔𝑒𝑛−𝑗}∗ (𝛾₁) (A1.3.3)

𝛾₁= ∑^𝑛_𝑖=1(𝑖 + (2^𝑗− 2)𝑛) ∗ (𝑁 − (𝑖 + (2^𝑗− 2)𝑛)) 𝛾₁= ∫ (𝑙 + (2₀^{𝑛 𝑗}− 2)𝑛) ∗ (𝑁 − (𝑙 + (2^𝑗− 2)𝑛))

𝛾₁= ∫ (𝑙𝑁 − 𝑙₀^{𝑛 2}− 2(2^𝑗− 2)𝑛 𝑙 + (2^𝑗− 2)𝑛 𝑁 − (2^𝑗− 2)²𝑛²) 𝑑𝑙

𝛾₁= 𝑛²𝑁 (¹

2+ (2^𝑗− 2)) + 𝑛³(−¹

3− (2^𝑗− 2) − (2^𝑗− 2)²) 𝛾₁= 𝑛²𝑁 ((2^𝑗) −³

2) − 𝑛³((2^𝑗)²− 3(2^𝑗) +⁷

3) (A1.3.4)

The result of Eq. A1.3.4 is added to Eq. A1.3.3:

𝑔 =_𝑁⁶₃∑^𝑔𝑒𝑛_𝑗=13 ∗ 2^{𝑔𝑒𝑛−𝑗}∗ 𝑛²𝑁 ((2^𝑗) −³₂) − 𝑛³((2^𝑗)²− 3(2^𝑗) +⁷₃)

𝑔 = ⁶

𝑁³∑^𝑔𝑒𝑛_𝑗=1(𝑛²𝑁 (3(2^𝑔𝑒𝑛) −⁹₂(2^{𝑔𝑒𝑛−𝑗})) − 𝑛³(3(2^{𝑔𝑒𝑛+𝑗}) − 9(2^𝑔𝑒𝑛) + 7(2^{𝑔𝑒𝑛−𝑗}))) It is not possible to change the summation into an integral because the variable 𝑎 is not large enough. Therefore the following summation expressions have to be used:

∑^𝑣_𝑢=11= 𝑣

∑^𝑣_𝑢=12^𝑢 = 2(2^𝑣− 1)

∑^𝑣_𝑢=12^−𝑢= 1 − 2^−𝑣

With the help of these expressions and Eq. A1.3.2, branching ratio 𝑔 in terms of generation number 𝑔𝑒𝑛 is obtained:

𝑔 =_𝑁⁶₃(𝑛²𝑁 (3(2^𝑔𝑒𝑛)𝑔𝑒𝑛 −⁹₂(2^𝑔𝑒𝑛) +⁹₂) − 𝑛³(−9(2^𝑔𝑒𝑛)𝑔𝑒𝑛 + 2^𝑔𝑒𝑛+ 3(2^{2𝑔𝑒𝑛+1}) − 7))

𝑔 =_𝑁⁶₃(_{3(2𝑔𝑒𝑛}^𝑁₋₁₎)³((3(2^𝑔𝑒𝑛)𝑔𝑒𝑛 −⁹₂(2^𝑔𝑒𝑛) +⁹₂ ) (3(2^𝑔𝑒𝑛− 1)) − (−9(2^𝑔𝑒𝑛)𝑔𝑒𝑛 + 2^𝑔𝑒𝑛+ 3(2^{2𝑔𝑒𝑛+1}) − 7))

𝑔 = ⁶

(3(2^𝑔𝑒𝑛−1))³((2^𝑔𝑒𝑛)²(9𝑔𝑒𝑛 −³⁹

2) + 2^𝑔𝑒𝑛(26) −¹³₂) (A1.3.5) To derive the final 𝑔 relation in terms of number of branch points 𝑏 the relation between 𝑔𝑒𝑛 and 𝑏 is required:

𝑏 = 3(2^{𝑔𝑒𝑛−1}− 1) 2^𝑔𝑒𝑛=^2𝑏+4₃

52 𝑔 =^(𝑏2(24𝑔𝑒𝑛−52)+𝑏(96𝑔𝑒𝑛−104)+(96𝑔𝑒𝑛−39))

(2𝑏+1)³

𝑔𝑒𝑛 =^𝑙𝑛(

2𝑏+4 3 )

𝑙𝑛(2) =𝑙𝑛(𝑏+2)−𝑙𝑛(3) 𝑙𝑛(2) + 1 𝑔 =^𝑏

2(24𝑙𝑛(𝑏+2)−𝑙𝑛(3)

𝑙𝑛(2) −28)+𝑏(96𝑙𝑛(𝑏+2)−𝑙𝑛(3)

𝑙𝑛(2) −8)+(96𝑙𝑛(𝑏+2)−𝑙𝑛(3) 𝑙𝑛(2) +57)

(2𝑏+1)³ (A1.3.6)

Equation A1.3.6 can be reduced into:

𝑔 =^𝑏2(34.62 𝑙𝑛(𝑏+2)−66.04)+𝑏(138.5 𝑙𝑛(𝑏+2)−160.2)+(138.5 𝑙𝑛(𝑏+2)−95.16)

(2𝑏+1)³ (A1.3.7)

The results of the branching ratio 𝑔 for star, comb and cayley tree shaped polymers are shown in Appendix 2, where simulations of freely jointed polymers are used to validate the obtained expressions.

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In document Eindhoven University of Technology MASTER Structural modification of LDPE during high-temperature processing Simons, Jérôme (pagina 46-54)