Characterization of Molar Mass

In Size Exclusion Chromatography (SEC) the differences in size of polymer molecules in dilute solutions are used to separate large molecules from small molecules. Here a column is filled with porous particles (the stationary phase) and subsequently filled with a liquid (the mobile phase).

When the polymer solution is added to the column, the small polymers are able to enter the pores and penetrate deep, while the large polymers that are larger than the pore size are not able to enter the pores at all. Therefore large polymers exit the column first and the small polymers last. This mechanism is named steric exclusion, where certain pore volume fractions are accessible for certain polymer sizes. Eq. 1.10 shows the relation between elution volume 𝑉_𝑒, solvent volume outside the pores 𝑉₀, solvent volume inside the pores 𝑉_𝑖 and distribution coefficient 𝐾_𝑑.¹¹

𝑉_𝑒= 𝑉₀+ 𝐾_𝑑𝑉_𝑖 (1.10)

10 The distribution coefficient 𝐾𝑑 is dependent on the size of the polymer chains and the used porous particles. Polymers that are larger than the largest pore size leave the column at elution volume 𝑉0

and polymers that are able permeate into all pores leave the column at elution volume 𝑉𝑖. The relation between elution volume and polymer size is shown in figure 1.4.

Ideally the distribution coefficient 𝐾_𝑑 is directly related with the size and even molar mass of the polymers. This, however, has proven to be difficult because there are multiple separation mechanisms occurring during size exclusion chromatography.¹¹ To obtain the absolute mass a calibration curve is sometimes used where the masses of known polymers are used. The calibration curve provides an easy way to obtain molar masses, but this method has a major pitfall. SEC only separates particles based on their hydrodynamic size and this size is, besides being dependent on the molar mass, dependent on the degree of branching and structure. Branching for example decreases the size of the polymers, resulting in a possible underestimated molar mass when using calibration curve. This is why additional analytical equipment in combination with SEC is needed to accurately determine the absolute mass of polymers.

1.2.1.2 Light Scattering

One of the few methods to for obtaining the absolute mass of polymers is light scattering. In light scattering, the oscillating field of the electric part of light creates dipoles from neutral molecules.

These polarized molecules are now able to affect the trajectory of incident radiation, resulting in a scattered beam. This event can take place multiple times, which impacts the intensity of the radiation beam. The decrease of intensity can be related with the mass, the Zimm formula is typically used:¹²

𝐾𝑐

𝑅_𝜃=_{𝑀𝑃(𝜃)}¹ + 2𝐴_𝑐𝑐 + ⋯ (1.11)

𝐾 is the optical constant, 𝑐 the concentration of the particles, 𝑅𝜃 the excess Rayleigh ratio, 𝑀 the molar mass, 𝑃(𝜃) the particle scattering function and 𝐴2 the second viral coefficient. The excess Rayleigh ratio 𝑅_𝜃 is dependent on the intensity of the scattered light beam (Eq. 1.12). The particle scattering function 𝑃(𝜃) is dependent on the shape of the particle, but 𝑃(𝜃) can also be

Polym er Siz e

Elution Volume 𝑉

_𝑒

𝑉 _𝑖 𝑉 ₀

Figure 1.4 – Hypothetical elution curve: elution volume 𝑽𝒆 versus polymer size. 𝑽𝟎 is the solvent volume outside the pores, 𝑽𝒊 the solvent volume inside the pores.

11 approximated with Eq. 1.13, for 𝑞²𝑅_𝑔² < 0.3.¹³ 𝑅_𝑔 is the radius of gyration and the scattering vector 𝑞 is dependent on the angle and wavelength of the incident light (Eq. 1.13).¹¹

𝑅_𝜃 =^𝑟2

𝑉

𝐼_𝜃−𝐼_{𝜃,𝑠𝑜𝑙𝑣𝑒𝑛𝑡}

𝐼₀ (1.12)

𝑃(𝜃) = 1 +^𝑞2₃^𝑅𝑔2 (1.13)

𝑞 =^4𝜋

𝜆 sin (^𝜃

2) (1.14)

𝐼_𝜃, 𝐼𝜃,𝑠𝑜𝑙𝑣𝑒𝑛𝑡 and 𝐼0 are the intensities of the solution, solvent and the used light, 𝑟 is the distance between the solution and detector and 𝑉 is the solution volume. 𝜆 and 𝜃 are the wavelength in a solvent and angle of the used light beam.

With equation 1.11 the molar mass of polymer particles can be determined by static light scattering.

During static light scattering, the light intensity fluctuations are averaged over time. The averages are determined for various angles and concentrations and subsequently can be used to construct a Zimm-plot where the molar mass is obtained by doing extrapolations for 𝑐 → 0 and 𝑞 → 0. A hypothetical Zimm-plot is shown in figure 1.5. The spacing parameter 𝑘 is chosen arbitrarily to simplify the extrapolation process.

The Zimm-plot however is only useful in ‘batch mode’, where concentrations can be varied. In 3D-GPC experiments the elution volumes from the SEC equipment are analyzed by light scattering online, where the concentration is fixed and typically low. For these measurements high-intensity laser lights are applied to give reliable results. Low angle laser light scattering (LALLS), used to measure scattering at a very low angle (at ~7 degrees), or multiple angle laser light scattering (MALLS), where multiple angles are used (~15-160 degrees), measure the light scattering and facilitate the determination of the absolute molar mass.

12 1.2.2 Characterization of Architecture

The absolute molar mass of polymers is found by combing SEC and LS. However, obtaining detailed information about the branching or structure is more complicated, because there is no direct method available to determine this. In this project intrinsic viscosity and rheological measurements are used to estimate the degree of branching and structure.

1.2.2.1 Dilute Solution Viscosity

When adding polymers to a solution the viscosity of solution increases. Here the hydrodynamic volume that is occupied by the polymer chain increases the viscosity. Specific viscosity 𝜂𝑠𝑝 is used to define the difference between the viscosity with and without the polymer (Eq. 1.15).

𝜂_𝑠𝑝=^{𝜂−𝜂𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛}

𝜂_{𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛} (1.15)

The specific viscosity can be measured by determining the flow rate in glass capillaries or by a differential viscometer.¹⁴ If the concentration of these polymers in solution is low enough, polymers are assumed to be isolated and therefore intermolecular polymer-polymer interactions are typically neglected. From here the limit 𝑐 → 0 can be taken to calculate the intrinsic viscosity [𝜂] of the polymer chain (Eq. 1.16).

[𝜂] = lim_𝑐→0𝜂_𝑠𝑝 (1.16)

For linear polymers, the Mark-Houwink equation is used to relate the intrinsic viscosity with molar mass (Eq. 1.17). 𝑀𝑣 is the viscosity average molar mass and 𝐾 and 𝛼 are constants that depend on

Figure 1.5 – Hypothetical Zimm-plot. The molar mass, radius of gyration and second viral coefficient are determined for various angles and concentrations and their quantity is obtained by doing extrapolations for c→0 and q→0.

13

[𝜂] = 𝐾𝑀_𝑣^𝛼 (1.17)

Both hydrodynamic volume and intrinsic viscosity are dependent on the degree of polymerization, degree of branching, structure and polymer-solvent interaction. The hydrodynamic volume is typically defined in terms of radius of gyration (Eq. 1.18). The mean square radius of gyration is the average square distance between monomer 𝑖 and all other monomers. The average distance is required if the polymer chain fluctuates in size over time (for instance because of Brownian motion).

< 𝑅_𝑔²> = ¹

2𝑁²∑^𝑁_𝑖=1∑^𝑁_𝑗=1< (𝑅_𝑖− 𝑅_𝑗)²> (1.18)

For polymers with the same molar mass, linear polymers have a larger hydrodynamic volume than highly branched and compact polymers, resulting in a larger contribution to the viscosity. Branching ratios 𝑔 and 𝑔′ are used to compare branched chains with its linear equivalent for both hydrodynamic volume and intrinsic viscosity:¹²

𝑔 =^<𝑅_<𝑅^{𝑔2>𝑏𝑟𝑎𝑛𝑐ℎ𝑒𝑑} relationship. Now 𝑔′ can be calculated directly. When evaluating 𝑔 and 𝑔′ for polymers with specific architecture, the “shielding effect” has to be taken in account.

If we consider a polymer in dilute conditions, the chain segments at the edges of the polymer particle alter the flow of the solvent near the chain segments at the center in such a way that these are partially shielded from hydrodynamic interaction with the solvent flow outside the particle. For large polymers, the contribution of the center segments becomes negligibly small to the resistance to the flow of solvent outside the particle.^15,16 Additionally, the amount of branching and the architecture of the polymers affect this contribution.

To include this effect in the evaluation of the two branching ratios, Zimm and Stockmayer defined the structure factor 𝜀:¹²

𝑔^′= 𝑔^𝜀 (1.21)

This structure factor is derived to be 𝜀 = 0.5 for non-free drained star branched polymers and estimated to be 𝜀 = 1.5 for comb shaped polymers, where the branches are small relative to the backbone.¹⁷ For autoclave and tubular LDPE, the values 𝜀 = 1.2 and = 2.0 , respectively, have been found by Kuhn.¹⁸ Scholte et al. report values of 𝜀 = 0.8 − 1.0 for low-density polyethylene.¹⁹ Tackx et al. determined the 𝑔 factor with SEC-MALLS measurements and found 𝜀 to decrease with increasing molar weight. The structure factors varied for one size autoclave and tubular LDPE between 𝜀 = 1.0 − 1.5 and 𝜀 = 1.2 − 1.8. In general a value for 𝜀 is chosen between 0.5 and 2.0.

𝑔 can be calculated by using relationships derived by Zimm and Stockmayer.²⁰ The first relation is valid for polydisperse trifunctional randomly branched polymers.

14 molecule. For monodisperse trifunctional randomly branched polymers the next relation can be used:

In this relation the average 𝑔 per molecule is described in the average branch points per molecule 𝑚. In addition, Zimm and Stockmayer give a relation to find the structure factor when using the relation mentioned above:

𝑔^′= 𝑔^𝜀 = 𝑔^2−𝛼 (1.24)

Here 𝛼 is the exponent in the semi-empirical law [𝜂] = 𝐾𝑀^𝛼 for linear polymers. However, Zimm and Stockmayer note “it is still hazardous to draw inferences about branching from empirical viscosity-molecular weight relationships”.²⁰

1.2.2.2 Rheology

An additional way to find more information about the structure of polymers is by conducting rheological measurements. Rheology is used to describe the flow behavior of materials under stress and it is shown that rheological measurement are able to reveal the structure and the degree of branching of polymers.^21,22 Introducing even low amounts of long chain branches to linear polymers has a significant effect on the rheological properties.

To measure rheological behavior an oscillatory rheometer is typically used. Here a sinusoidal shear deformation is applied to the sample from which the stress response is measured. In figure 1.6, a rheometer is schematically shown, where the bottom plate oscillates with a certain frequency and the top plate measures the stress response. . Because of the rotating bottom plate, the strain is time dependent (Eq. 1.25). In this equation 𝜔 is the frequency of the strain and 𝑡 the time. The stress response can be proportional to the strain or, for most polymers, can lag behind. Here Eq. 1.26 is used, where 𝛿 is the phase lag.

𝜀(𝑡) = 𝜀₀sin(𝜔𝑡) (1.25)

𝜎(𝑡) = 𝜎₀sin(𝜔𝑡 + 𝛿) (1.26)

When a material is ideally elastic, the stress response is proportional to the applied strain (figure 1.6). This behavior is modeled by using a Hookean spring, where the applied stress 𝜎 is equal to the strain 𝜀 times the spring (or elastic) constant 𝐸. Equation 1.27, similar to Hooke’s law, is used to model the elastic behavior. The phase lag 𝛿 can be found by inserting Eq. 1.25 and Eq. 1.26 to 1.27.

For an ideally elastic material the phase lag is equal to zero, because the stress is proportional to the strain.

𝜎(𝑡) = 𝜀(𝑡)𝐸 (1.27)

15 The stress response of an ideally viscous material is proportional to the applied strain rate (figure 1.6). A Newtonian dashpot model can be used to describe the viscous response. Eq. 1.28 is used here, where the applied force 𝜎 is equal to the strain rate 𝑑𝜀 𝑑𝑡⁄ times the viscosity. Here the phase lag is shifted one fourth of a period (𝛿 =^𝜋₂).

𝜎(𝑡) = 𝜂^{𝑑𝜀(𝑡)}_𝑑𝑡 (1.28)

Most polymers are viscoelastic, showing both viscous (dashpot) and elastic (spring) behavior. The phase lag has therefore a value between 0 and ^𝜋₂. An example of a simple model for viscoelastic behavior is the Maxwell model. In a Maxwell model, the spring and dashpot are connected in series.

The overall strain rate equation results in: ^𝑑𝜀_𝑑𝑡 =^𝑑𝜀_𝑑𝑡|

𝑆+^𝑑𝜀_𝑑𝑡|

𝐷=¹_𝐸^𝑑𝜎_𝑑𝑡+^𝜎_𝜂. Dynamic modulus 𝐺 can be derived from this equation, that is the ratio of stress 𝜎 over strain 𝜀. This modulus can be split into two elements, storage modulus 𝐺′ and loss modulus 𝐺′′, that represent the elastic and viscous behavior of the viscoelastic material, respectively.¹⁰ Rheometer measurements typically give the storage and loss moduli.

In these measurements, it is assumed that the applied strain scales linearly with the stress response.

Therefore the dynamic moduli are independent of the applied strain if this assumption holds true.

When the strain amplitude is increased however, deformations, structural changes and phase transitions can occur that give different rheological behavior.²⁰ The strain does not scale linearly with the stress anymore, resulting in nonlinear rheological response. It is shown that nonlinear rheological measurements potentially can be used to provide information about the properties of

𝑅𝑜𝑡𝑎𝑡𝑒

Figure 1.6 – (Left) Schematic representation of a rheometer. The bottom cylinder is oscillating at with a certain frequency.

The material responds to these deformations and this is measured at the top cylinder. (Right) Typical response curves of elastic, viscous and viscoelastic materials. For elastic materials the stress response is proportional to the strain, for viscous material the stress response is proportional to the strain rate and for viscoelastic material the stress response lies in between the elastic and viscous stress response.

16 the materials, including branching and structure.^24,25 Blood vessels for example stiffen (or harden) when a large amount of strain is applied.²⁶ An example of this strain hardening for several biological gels is shown in figure 1.7.²⁶ Here can be seen that for lower strain values the dynamic moduli are independent of the strain, while a dependency occurs for higher strain values. Large Amplitude Oscillation Shear (LAOS) test reveal the nonlinear behavior of materials, where for example the extensional viscosity as a function of time is used. The transient extensional viscosity is the ratio between the extensional stress and the strain rate:

𝜂_𝐸⁺= ^{𝜎𝑒𝑥𝑡}

𝑑𝜀/𝑑𝑡 (1.29)

In experiments, however, it might be difficult to measure the steady state extensional viscosity accurately. Therefore 𝜂_𝑒⁺ is typically compared with linear viscoelastic behavior via the Boltzmann superposition principle:

𝜂_𝐸⁺= 3 ∫ 𝐺(𝑡)𝑑𝑡₀^𝑡 = 3𝜂⁺ (1.30)

For a low 𝑡 the extensional viscosity 𝜂_𝐸⁺ over steady state shear viscosity 𝜂⁺ is equal to 3 (also named the Trouton ratio). With the empirical Cox-Merz rule the steady state shear viscosity is linked to the complex viscosity, when shear rate 𝛾̇ is equal to frequency 𝜔:^29,30

𝜂⁺(𝛾̇) = 𝜂^∗(𝜔) (1.31)

The complex viscosity 𝜂^∗ is obtained from dynamic moduli 𝐺^′ and 𝐺^′′:

𝜂^∗=^{√𝐺′(𝜔)2}_𝜔^{+𝐺′′(𝜔)2} (1.32)

For higher 𝑡 values the extensional viscosity can deviate from the linear viscoelastic behavior and is an indicator for certain structural characteristics of the polymer sample. In our case highly branched LDPE is used where strain hardening occurs during extensional viscosity measurements. Here the applied strain strongly orientates the polymer chains that increases the extensional viscosity and stiffens the material. In film blowing for example strain hardening is crucial for stabilizing the resin.

Figure 1.7B shows a hypothetical plot for strain hardening compared with the linear viscoelastic result.

Extensional Viscosity 𝜂𝐸+

Time t

Linear Viscoelastic Behavior Nonlinear Viscoelastic Behavior

Figure 1.7 – (Left) Nonlinear response of various biological materials. Figure obtained from [23]. (Right) Hypothetical transient extensional viscosity curve for a material showing nonlinear viscoelastic behavior (red). The dashed blue line represents the material if it would show linear viscoelastic behavior. The line is obtained from the dynamic moduli and by using equations 1.30-1.32.

17 1.2.3 The “Branch-on-Branch” (BoB) Model

Rheological measurements give valuable information about polymer samples, but it is hard to extract detailed structure characteristics. In the models of this project the “Branch-on-Branch” (BoB) model of Das et al.⁴ is incorporated to link molecular characteristics to viscoelastic responses. In this BoB model theoretical understanding of polymer dynamics is used to predict the viscoelastic response of polymer melts. The model is specifically suited for estimating the response of highly branched polymers.

Branched polymers differ in motion with respect to linear polymers because of the presence of branches. A common example is the motion of a 3-armed star, shown in figure 1.8. In a polymer melt the star is entangled with other polymers that restrict the movement of the star. The remaining space the star is allowed to move in has a tube-like shape, as shown in figure 1.8. The tube model, developed by de Gennes³¹, is used to define the motion of polymers in the space the polymer is allowed to move in. Linear polymers in this tube model would be able to move through the tube with a snake like motion, termed reptation. Adding a branch to the polymer restricts the movement:

the side-arm has to be dragged into to tube section where the polymer is moving into, which entropically unfavorable. This slows down the stress relaxation of the star polymer.

For polymers with more than one branch the relaxation becomes more difficult. Short polymer segments that have a free end relax faster than large segments that are on both sides connected to other segments. To describe the relaxation order of branched polymer hierarchical relaxation is used. First all segments with a free end relax until all segments are relaxed. Next the segments that are connected to these segments are allowed to relax but with added drag that is caused by the relaxed segments. This relaxation procedure is repeated until the whole polymer is relaxed. Here the quantity seniority is used here to describe the hierarchy of relaxation. This quantity is determined by finding the number of segments to the furthers free end on both sides of segment 𝑥. The smallest value of these two is the seniority. Segments with a free end have a seniority of 1 and relax first, the segments connected to these have a seniority of 2 and relax second, et cetera. With this value the

‘deepness’ of the segment in the polymer is specified. Besides hierarchical relaxation additional effects like primitive path fluctuations and constraint release are added to this theoretical frame from which ultimately the linear viscosity behavior is predicted.³²

Figure 1.8 – In a polymer melt the star is entangled with other polymers that restrict the movement of the star.

The blue dots represent the topological constrains set by the polymer melt. The star is allowed to move inside the tube represented by the dashed black lines. If the star polymer leaves the tube, the side-arm has to be dragged into to tube section where the polymer is moving into.

18 Nonlinear viscoelastic behavior is modelled by using the Pom-Pom model.³³ A pom-pom polymer consists of a backbone that is connected on both sides with 𝑦 arms that all have a free end. With the pom-pom model the orientation and stretch of the backbone segment is described from which the strain hardening can be estimated. The value priority is used to account for the maximum stretch of segment 𝑥. The number of free ends that lie at the outer ends of the polymer are counted for both sides of segment 𝑥, from which the smallest value of the two is taken as the priority. The priority and seniority values can be used as an indicator for the structure of the polymer.

1.2.3.1 BoB Model Input

In order to use the BoB model, a couple of parameters have to be known a priori: the dynamic dilation exponent 𝛼, the number of polymer units in a entanglement 𝑁𝑒, equilibrium time 𝜏𝑒, the molar mass of a polymer unit 𝑀0, the temperature 𝑇 and the polymer melt density 𝜌. The dynamic dilation exponent 𝛼 is typically varied between 1 and 4/3. 𝑁𝑒 is obtained from the entanglement molar mass 𝑀𝑒 which can be determined from 𝑀𝑒=^𝜌𝑅𝑇_𝐺

𝑁0 or 𝑀𝑒=⁴₅^𝜌𝑅𝑇_𝐺

𝑁0, where 𝐺_𝑁⁰ is the plateau modulus.^34,35 For the equilibrium time 𝜏_𝑒= ^{𝜁𝑎2𝑀𝑒}

3𝜋²𝑘𝑇𝑀0 can be used, where 𝜁 is the monomeric friction coefficient and 𝑎 the tube diameter.³² 𝑀0 is 28 g/mol for LDPE. In our model the values used by Read et al.³⁶ will be used (Table 1).

The polymers that are used in the BoB model are represented by a matrix that is included in the polyconf file. This polyconf notation is the following. The first line is used for segment 0 that has two possible segments or free ends on each side. If segment 𝑥 is connected to segment 0 the value 𝑥 is added to the first line. If segment contains a free end, the value −1 is added to the line. For a polymer with 2 branches, the polyconf file will look like:

The first four columns are used for the connectivity specification. The fifth and sixth columns are

The first four columns are used for the connectivity specification. The fifth and sixth columns are

In document Eindhoven University of Technology MASTER Structural modification of LDPE during high-temperature processing Simons, Jérôme (pagina 10-0)