Examensarbete 30 hp Juli 2020
Negative energy elasticity and a model for the behavior of the residual strain in doubly cross-linked gels fabricated by shear strain
Therese You
Populärvetenskaplig sammanfattning
Polymergeler är material som är av stort intresse för det medicinska fältet. Speciellt är det av intresse inom fält som vävnadsrekonstruktion av mjuka vävnader. Det är svårt att hitta syntetiskt material som kan ersätta till exempel brosk eller muskler. För detta behöver du material som din kropp inte avvisar, som beter sig som det som ersätts och tillåter exempelvis blodkärl eller celler att överleva. Detta är svårare än man skulle tro då kroppen är väldigt känslig för vad som implanteras. En av orsakerna till att det är svårt att hitta passande geler är för att de ofta inte är så tåliga. Geler är generellt ganska sköra, medan det vi behöver är relativt tåliga material så de inte går sönder i kroppen eller har andra negativa biverkningar. Hydrogeler är en typ av polymergel som består till större del av vatten, ibland över 90%. Detta är bra för att vi vet att kroppen inte har något emot vatten, dock blir de istället ganska ömtåliga. Därför har mycket forskning riktat sig mot att hitta tekniker som kan skapa starkare hydrogeler.
En av orsakerna till skörheten är att polymergel består av ett nätverk av polymerkedjor.
Dessa hamnar ofta som spagetti på en tallrik, lite slumpmässigt. Det gör att allt håller ihop i en klump, men att en del av gelen kanske är svagare än en annan del beroende på hur kedjorna är bundna just där. Detta är inte bra för att om vi ska kunna använda ett material är det viktigt att kunna förutse hur det beter sig. Men om alla kedjor binder slumpmässigt är det väldigt svårt att spå gelens egenskaper. Därför har man tittat på geler där man kan kontrollera hur kedjorna binder sig till varandra. Sådana mer homogena geler är svåra att hitta och kan vara svåra att tillverka. Man spår dock att en gel utan intrasslade kedjor med kontrollerad struktur skulle kunna producera en gel som är stark och tålig. Kontroll över strukturen, hur kedjorna hamnar, blir viktigt också om man vill styra mekaniska egenskaper eller vart exempelvis celler och blodkärl ska hamna. Geler har många spännande egenskaper som gör de intressanta, exempelvis deras elasticitet.
Det finns i huvudsakligen två bidrag till att geler är elastiska och tänjbara. En är ett energetiskt bidrag till elasticiteten som mer omfattar interaktionen mellan kedjor och energin som sparas i gelen när den deformeras. Det andra är ett entropiskt bidrag som mer omfattar hur många olika former eller strukturer de olika kedjorna kan ta. För många polymergeler så brukar man anta att energibidraget är försumbart mot entropibidraget.
Ofta så approximerar man att energibidraget är noll.
En grupp forskare har tillverkat en typ av gel som kallas Tetra-PEG gel, gjorda från fyrarmad poly(etylenglykol). Dessa geler består av kedjor som ser ut som plustecken där en kedja har fyra armar. På grund av sättet nätverken byggs för denna typ av gel har man mycket mer kontroll över strukturen än för många andra geler. Ett plustecken kan bara binda sig till ett annat plustecken med rätt kemi. På detta sättet skapar du ett nätverk av kryss. Du kan därmed skapa en nästan ideal gel. För dessa Tetra-PEG geler har det visat sig att energibidraget till elasticiteten är negativt och därmed inte försumbart som antaget för många andra geler. Det indikerar att det kan behövas titta över mycket forskning inom geler där man har antagit att bidraget är försumbart utan att ha dubbelkollat. Dessa geler har vidare bevisats vara biokompatibla och lätta att tillverka vilket är otroligt spännande för framtiden inom hydrogeler. Dock så kvarstår fortfarande att de är ganska sköra.
En teknik för att göra de starkare har visat sig att vara genom att kombinera två olika nätverksstrukturer i samma gel. Det är som att du har ett mönster av tråd som du kombinerar med ett annat mönster av tråd för att göra det mer tåligt och slitstarkt.
Principen är samma för dessa geler. De kallas för doubly cross-linked gels, dubbel- tvärbundna geler. Dessa typer av geler gjorda på Tetra-PEG gel har studerats tidigare och då har man kunnat bevisa att de är starkare än vanlig Tetra-PEG gel. Men man skulle ändå vilja titta mer på hur de dubbel-tvärbundna gelerna kan tillverkas och hur de beter sig under stor mekanisk belastning för att säkra sig att de är tåliga. Det är bland annat det som har studerats under detta examensarbete.
Dubbel-tvärbundna geler skapades med hjälp av skjuvspänning där du förvrider det naturliga nätverket hos Tetra-PEG geler och på det sättet introducerar ett sekundärt nätverk. Resultatet är en gel med två nätverksstrukturer. Det visade sig att dessa geler tillverkade med skjuvspänning hade samma skjuvmodul, som är ett mått på gelens elasticitet, oavsett hur mycket du förvred gelen under tillverkning. Sedan indikerade resultat att energibidraget var även signifikant negativt för dessa lite mer komplexa geler baserade på Tetra-PEG gel. Vilket skulle stärka det faktum att man kanske måste titta över tidigare forskning där man har trott att bidraget är försumbart. Sedan presenterades det en formel för spänning som kvarblev i gelen efter man slutade förvrida. Detta är nödvändigt så man kan förutspå hur spänningen sitter kvar i gelen beroende på hur mycket man förvrider den under tillverkning. Sist men inte minst så verkade även dessa dubbel-tvärbundna geler visa ökad elasticitet. Dock inte signifikant skillnad jämfört med vanlig Tetra-PEG gel. Däremot är det viktigt att ta alla dessa slutsatser med en nypa salt då mätningar allmänt bara kunde göras en gång för varje pålagd skjuvspänning. I vanliga fall vill man göra flera repetitioner av samma mätning för att bekräfta resultaten. Men det hanns inte med under detta examensarbete. Dock är resultaten fortfarande väldigt spännande och uppmuntrar till mer forskning på dubbel-tvärbundna geler baserade på Tetra-PEG gel. Vidare motiverar erhållna resultat till mer undersökning av energibidraget till elasticitet för andra polymergeler.
Contents
1 Introduction 7
1.1 Tetra-PEG gel . . . . 7
1.2 Temperature dependence of the elasticity of polymer gels . . . . 8
1.3 Doubly cross-linked gels based on Tetra-PEG gel . . . . 9
1.4 Effect of large applied strains on rubber-like gel . . . . 10
1.5 Project aim . . . . 10
2 Materials and Methods 10 2.1 Tetra-PEG gel . . . . 10
2.2 Gelation process of Tetra-PEG gel . . . . 11
2.3 Theoretical method background . . . . 11
2.4 Rheological testing method . . . . 13
2.5 Controlled parameters . . . . 15
2.5.1 Temperature, T . . . . 15
2.5.2 Molecular weight, M . . . . 15
2.5.3 Connectivity, p . . . . 16
2.5.4 Concentration, c . . . . 16
2.5.5 Applied strain, γi . . . . 16
3 Experimental 17 3.1 Equipment and material . . . . 17
3.1.1 Equipment . . . . 17
3.1.2 Material . . . . 17
3.2 Procedure . . . . 17
3.3 Measurement program . . . . 17
4 Results and discussion 18 4.1 Conclusions from compression testing on Tetra-PEG gel . . . . 18
4.2 Rheological measurements on doubly cross-linked gel . . . . 18
4.2.1 Shear modulus for the first network structure . . . . 19
4.2.2 Stress dependence of the applied shear strain . . . . 21
4.2.3 Study of the residual strain . . . . 22
4.2.4 Temperature dependence of the shear modulus . . . . 24
4.2.5 Sources of error . . . . 27
5 Conclusion 28
6 Future work 28
7 Acknowledgments 29
8 References 30
Appendices 31
A Compression testing of regular Tetra-PEG gel 31
A.1 Compression testing method . . . . 31
A.1.1 Controlled parameters . . . . 32
A.2 Experimental . . . . 33
A.2.1 Equipment and material . . . . 33
A.2.2 Procedure . . . . 34
A.2.3 Testing method . . . . 34
A.3 Results and discussion . . . . 35
A.3.1 Method development: Method 1 . . . . 35
A.3.2 Testing results for method 1 . . . . 35
A.3.3 Method development: Method 2 . . . . 38
A.3.4 Testing results for method 2 . . . . 39
A.3.5 Temperature control of the sample holder . . . . 40
A.3.6 Method development: Method 3 . . . . 41
A.3.7 Testing results for method 3 . . . . 41
A.3.8 Investigating factors affecting the compression tests . . . . 43
A.3.9 Shrinkage test of gel in oil . . . . 45
A.4 Conclusions for compression testing of Tetra-PEG gel . . . . 47
B Data from rheological measurements for the doubly cross-linked gels 48 B.1 Experimental values for the shear stress τ . . . . 48
B.2 Exponential fitting of apparent shear modulus . . . . 48
1 Introduction
The general purpose of this master thesis was to study polymer gels and their mechanical properties. More specifically it was to study doubly cross-linked gels based on tetra- poly(ethylene glycol) (Tetra-PEG). Polymer gels in general are 3D networks of polymers that are swollen in solvent [1]. If the main solvent is water these polymer gels are usually referred to as hydrogels. Hydrogels offers many new opportunities within fields such as tissue engineering [2]. They usually have a high water content and an elasticity that often mimics the physical properties of many human tissues. PEG gels in particular are proven bio-compatible and have shown to support cell migration and proliferation in vitro. Additionally, PEG-gels are hydrophilic, have low non-specific adsorption of protein and do not degrade by mammalian enzymes. However to widen the use of hydrogels in the medical field it is necessary to be able to control and improve both the mechanical properties and structure. There has been several important discoveries in recent years such as interpenetrating polymer networks, self-assembly techniques and co-polymerization [3]. However poor mechanical properties is still a limiting factor for the use of a lot of hydrogels. There has been a lot of approaches to control and improve mechanical properties for polymer gels, such as raised elasticity or rigidity, without introducing toxicity or other inconveniences. One of the methods is to introduce interpenetrating networks to the structure. These are more complex polymer gels like doubly cross- linked gels, double-network rubbers and double-network gels [4]. The introduction of a second network allows for mechanical and optical anisotropies. The creation of synthetic anisotropic mechanical materials is of importance for medical application as a second network can provide improved physical properties, such as raised elastic modulus, and mimic complex behavior of for example muscles.
Polymer networks are usually characterized by heterogeneity introduced by cross-linking needed by the gels to hold their shape. However it is very difficult to control or predict the different heterogeneities, their formation mechanism and how they relate to each other in a network [1]. Due to the complexity of heterogeneities in a polymer gel it is also difficult to understand the effect of them on the physical properties. There exist several models predicting different physical properties. However as the true structure is not known it is very difficult to accurately predict the properties. To be able to better understand the formation mechanism and the effect of heterogeneities it is in theory reasonable to start studying a homogeneous polymer network. A homogeneous polymer network could provide basic understanding on how to control mechanical properties and to understand the structure of the polymer networks in hydrogels.
1.1 Tetra-PEG gel
Gels based on tetra-poly(ethylene glycol) have received much attention due to their highly suppressed heterogeneity [1]. These gels will hereafter be referred to as Tetra-PEG gels. There have been several attempts to create homogeneous polymer networks with approaches like click chemistry and gelation by end-crosslinking [5][6]. These approaches have yielded less heterogeneous networks, but as the cross-links are introduced randomly they provide little control over the homogeneity. Tetra-PEG gels on the other hand achieve its suppressed heterogeneity thanks to a method called AB-type cross-link coupling. The method provides three key factors. The first key factor is that the method uses an end-
crosslinking system that allows for a lower degree of freedom for the network formation and a constant network strand length. The second factor is that it uses a diluent, which decreases the amount of entanglements of the polymers. Finally it allows for control of the reaction rate through the pH of the diluent used. If the reaction rate can be controlled the mixing can be controlled and allow for the reaction to happen more homogeneously. The reaction for Tetra-PEG gel happens between mutually reactive tetra-armed precursors with the same shape (figure 1). The diluent is aqueous buffer solution. Tetra-PEG gels created with AB-type cross-link coupling do have some heterogeneity, but is eminently more homogeneous than other polymer gels. They are also proven bio-compatible and are relatively easy to fabricate.
Figure 1: Illustration of two different precursors with end-groups R1 and R2 resulting in a Tetra-PEG gel
1.2 Temperature dependence of the elasticity of polymer gels
As mentioned earlier, near-homogeneous networks are very hard to come by. But they are expected to have high mechanical strength, such as large elasticity, and might help to understand the basics of polymer gel structures. This could provide a whole new understanding of polymer gels and their physical properties. One of the most basic mechanical properties of polymer gels is the elasticity. For polymer gels the elasticity has two contributions in the form of energy elasticity and entropic elasticity [7]. Elasticity is often described by Young’s modulus (E ), also known as the elastic modulus, or the shear modulus (G ), also known as the modulus of rigidity [8]. The difference between the two is that Young’s modulus is defined by the ratio of tensile stress (σ [Pa]) to tensile strain (ε [-]), meaning longitudinal stress to longitudinal strain. When measuring the Young’s modulus uni-axial tension or compression is measured:
E = σ ε
The shear modulus, G, on the other hand is defined as the ratio of shear stress (τ [Pa]) to shear strain (γ [-]). G is measured under shear deformation instead of uni-axial deformation:
G = τ γ
For isotropic materials the two moduli are connected by the relation:
E = 2G(1 + ν)
Where ν is known as Poisson’s ratio which describes the ratio of the relative contraction strain (laterial, radial or transverse) to the relative extension strain in the direction of the applied stretching force [9]. Tetra-PEG gels have approximately a Poisson’s ratio of
0.5 due to their rubber-like elastomeric behavior for small strains. This means that it is assumed that the gels behave as perfectly in-compressible isotropic material that is deformed elastically within small strains.
Classic rubber elasticity theory mainly focuses on the entropy contribution as small energy contributions to elasticity are observed for many rubber materials [7]. The theory has also been applied to polymer gels (rubber and solvent), but in a smaller scale. When studying Tetra-PEG gels it was found that the energy elasticity contribution of the shear modulus could be a significant negative value. The finding of the significant negative energy elasticity of Tetra-PEG gel indicate implications for past research on the elasticity of polymer gels. For example in previous studies of Tetra-PEG gels the energetic elasticity contribution was estimated to be zero, which is not a valid assumption according to the new finding. A suggested cause for this negative energy elasticity was solvent-polymer interaction. However to properly establish the negative energy elasticity, its origin and establish if the findings are universal in other polymer gels further research needs to be done. It therefore becomes of interest to study the energy elasticity of other polymer gels, such as more complex Tetra-PEG gels.
1.3 Doubly cross-linked gels based on Tetra-PEG gel
Doubly cross-linked gels are described as gels with covalently linked network structures [4].
The doubly cross-linked gel in this case has two network structures present in one polymer network. During this thesis the second network structure was introduced by applying a shear strain on a Tetra-PEG gel during the gelation reaction (figure 2). This created two different network structures in one network that coexisted with each other. Hence one can create a more complex network structure based on Tetra-PEG gel without introducing another material. To fabricate the doubly cross-linked gel one has to allow the first network structure to gelate and then during the gelation process introduce a mechanical field (in this case shear strain) that creates a deformed structure of the regular first network structure. This mechanical field is applied during the remaining of the gelation process. When removed the system would have two network structures, the first natural structure and the second induced structure.
Figure 2: Illustration of the fabrication of doubly cross-linked gel using shear strain Doubly cross-linked gels based on Tetra-PEG gels have been fabricated before by applying uni-axial stretching that resulted in enhanced Young’s modulus [4]. These results were
well explained based on classical two-network theory, where the elastic free energy (W) can be described as the sum of the energy from the first (W1) and second (W2) network structures [10][11][12]:
W = W1+ W2 (1)
Hence, the study quantitatively validated the two-network theory. The research also concluded that doubly cross-linked gels fabricated by uniaxial stretching provided enhanced Young’s modulus and residual strain compared to the Tetra-PEG gel it was based on.
However it is still of interest to confirm whether the energy contribution to the elasticity for this more complex structure is negative as proved for regular Tetra-PEG gel [7]. It is also of interest to study the behavior of mechanical properties, such as the shear modulus, when exposed to larger external mechanical fields during fabrication. This is important as more information of the behavior of the gels are needed for the possible future application of them within for example the medical field.
1.4 Effect of large applied strains on rubber-like gel
As the doubly cross-linked gels are fabricate by shear strain it is important to know about nonlinear effects that might affect the obtained results. Firstly due to stress relaxation nonlinear behavior at larger strains and nonlinear behavior at higher temperatures can be expected for certain polymer gels [13]. Furthermore it is known that larger applied strains create higher energy within the network due to the interaction between the chains. As the strain increases the conformity of the gel decreases and the interaction between chains increase. This could all affect large strain results and cause observed nonlinearity. In a study of nonlinear effects during stress relaxation tests utilizing rheology on poly(vinyl chloride) (PVC) gels it was found that the gel exhibited nonlinear damping of the shear modulus with increased strain [14]. In the concluding remarks the weak damping of the shear modulus G was attributed to changes in the fractal structure of the gel. These changes was due to strain-induced partial rupture of the crystal domains. Based on these findings it is possible that damping of the shear modulus might occur for large strains for Tetra-PEG gel as well.
1.5 Project aim
The aim of this master thesis was to investigate the energy elasticity and study the behavior of the modulus of rigidity for doubly cross-linked gels based on Tetra-PEG.
2 Materials and Methods
2.1 Tetra-PEG gel
The Tetra-PEG gels used during the experiments were made up of Tetra-PEG-MA (tetra- maleimide-terminated PEG) and Tetra-PEG-SH (tetra-thiol-terminated PEG) (figure 3).
The solvent used to make the gels was citrate-phosphate buffer (CPB) with a pH of 3.8 and ionic strength of 200mM. This particular combination of precursors and solvent was selected based on previous studies on Tetra-PEG gel [7].
Figure 3: Illustration of the two precursors and the resulting Tetra-PEG gel
2.2 Gelation process of Tetra-PEG gel
The gelation process of the regular Tetra-PEG gel consisted of diluting the two precursors separately in the aqueous buffer. The precursors were then mixed in calculated ratios with the help of a conditioning mixer. The conditioning mixer was a Thinky AR-100. This mixer allowed for programs to be made with a set time for mixing and a set time for deaeration of the sample (table 1) [15]. When mixing a precursor with diluent the mixing and deaeration time was longer. The shorter program was used when the mixing of the two precursors occurred. It is of importance to mix the two precursors thoroughly, but also efficiently as the gelation start as soon as the two precursors come in contact. The time from when the two precursors are mixed until testing can be done on the doubly cross-linked gels is 49 200s (≈ 14h). The gelation time is governed by the chosen pH of the diluent, which was chosen to ensure a relatively fast, but also nearly homogeneous gelation.
Table 1: Mixing programs Program Mixing Dearation
1 2min 1min
2 15s 15s
2.3 Theoretical method background
To experimentally determine the energy (GE) and entropic (GS) contributions to the elasticity one has to measure the stress, τ , as a function of temperature, T, for a constant sample volume [7]. The stress can then be separated into the energy (τE) and the entropic (τS) contributions. Considering an in-compressible elastomer applied in an isothermal process with an external strain, γ, the corresponding stress can be measured [16]. The stress is defined as the force applied (F [N]) divided by the cross sectional area (A [m2]) and the resulting unit is Pascal [Pa] [8]. The shear stress is denoted as τ and is the shear force in the plane of the area divided by the area. The strain is defined as the change in length or displacement divided by the initial length or initial position. For shear strain, γ, this is described as the horizontal displacement divided by the height of the displaced segment. Strain is therefore dimensionless.
The stress is dependent on both the temperature and strain meaning τ = τ (T, γ) for
rheological measurements [7]. This stress is related to Helmholtz free energy density, f = f (T, γ). Hence τ can be defined as τ = ∂f /∂γ. Based on thermodynamic laws f = e − T s, where e is the internal energy density and s is the entropy density and given that τ = τE + τS one can separate the two contributions from each other.
τ = τE+ τS gives τS = −T∂s
∂γ and τE= ∂e
∂γ
Where τ [Pa] is the stress and τE[Pa] and τS[Pa] are the energy and entropic contributions to the stress, respectively. T [K] is the temperature, γ [-] is the strain, e [J] is the internal energy density and s [J ×K−1] is the entropy density. Then according to the Maxwell relation:
∂s
∂γ = −∂τ
∂T We get
τS(T, γ) = T∂τ
∂T(T, γ) (2)
Then by utilizing equation 2 τScan be determined by measuring the temperature dependence of τ when γ is fixed. From the obtained results τE can be calculated from τ = τE+ τS
(figure 4 A).
Figure 4: Illustration of A) τE and τS estimated from measured τ and B) estimated T0 from a G-T plot
Based on earlier research it was found that the shear modulus G was nearly a linear function of the temperature T in the measured range of 5-25◦C for regular Tetra-PEG gel [7]. It was also found that the x-axis intercept (T0) in a G-T plot was independent of the connectivity p of the Tetra-PEG gel with precursors that had a molecular weight (M ) and concentration (c) (figure 4 B). The results seemed to indicate that G could be described according to:
G(T, M, c, p) = a(M, c, p)[T − T0(M, c)]
Where the shear modulus G [Pa] can be described by a pre-factor called a [Pa/K], the measured temperature T [K] and the x-axis intercept T0 [K] in a G-T plot.
The earlier findings showed that the entropy contribution to the elasticity could be described as GS = aT and that the energy contribution to the elasticity could be described as GE = aT0. If a negative value for GE was to be indicated for the doubly cross-linked
gels the earlier found results could be validated for more complex Tetra-PEG gels and indicate some universality to the results of negative energy elasticity.
2.4 Rheological testing method
To study the doubly cross-linked gels rheological measurements were used. Rheometery provides a simple way to introduce the shear strain γ needed to form the second network structure, while continuously studying the samples. The dynamic shear rheometers used were of the models Anton Paar Physica Modular Compact Rheometer 301 and Anton Paar Modular Compact Rheometer 302. These were the same rheometers used when the negative energy elasticity of the normal Tetra-PEG gel was observed [7]. For the measurements Anton Paar concentric cylinders (CC17) were used. The measuring probe was also sealed with a silicone barrier with a wet tissue placed inside surrounding the cylinder to ensure minimum drying of the gel when measuring the samples (figure 5).
Figure 5: Image of precautions taken to minimize sample drying
Earlier experiments on doubly cross-linked gels based on Tetra-PEG gel fabricated by uniaxial stretching validated the two-network theory [4]. Meaning that the elastic free energy could be described as the sum of the first and second network structures for doubly cross-linked gels (equation 1). It is of interest to also confirm if doubly cross-linked gels fabricated by shear strain can be described by the two-network theory. The equation for elastic free energy (W) can be applied to doubly cross-linked gel and is described by classical rubber theory according to:
W = G
2(λ2x+ λ2y+ λ2z− 3)
Where G is the shear modulus and λj (j = x, y, z) is the displacement ratio of the network structure relative to a network at ease. For doubly cross-linked gel fabricated by shear strain the formula was derived to be [17]:
W = G
2(γ2− 3)
The doubly cross-linked gels that were studied essentially had three reference states. Each state had its own formula for the elastic free energy for the first (W1) and second (W2) network structure (figure 6). The first state considered the un-deformed regular Tetra- PEG gel. Here the elastic free energy is 0 for both network structures. Any tests done at this state would equal tests done on the regular Tetra-PEG gel. The second state was when applying the shear strain. At this point the regular Tetra-PEG gel had been allowed to gelate for a while and then the applied strain, γi, was introduced. Finally the
applied strain was removed and only residual strain, γS, was remnant in the system. The states are visually represented and following formulas for the elastic free energy can be seen below (figure 6).
Figure 6: Illustration of the reference states for the rheological measurements of the doubly cross-linked gel where you have the un-deformed state, the state with applied shear strain and finally the residual strain after removing the applied shear strain
It was of interest to study the residual strain dependent on the applied strain to see how the gel behaved depending on the magnitude of the applied strain. A formula for the residual strain (γS) can be derived from the elastic free energy of the last reference state mentioned above. The formula for the elastic free energy for the two networks structures at this point is:
W1 = G1
2 (γS− γ)2 W2 = G2
2 (γS− γi− γ)2
Where G1 refers to the virtual shear modulus of the first network structure and G2 for the second network structure, when independently formed in un-deformed states. The shear modulus G1 is approximated by the value of G at the end of the gelation of the regular Tetra-PEG gel before applying any shear strain and G2 is approximated by subtracting G1 from the value of the measured G after the applied shear strain is removed and the gelation is complete (figure 7). For completed experiments the value of G was approximated by the observed storage modulus (G’) as the loss modulus (G”) was approximately zero during the measurements of the gels. Meaning G for the doubly cross-linked gel is approximated by the sum of G1 and G2.
Figure 7: Illustration of G1 and G2 approximated from a G-t plot for doubly cross-linked gels
The stress-strain relationship for the doubly cross-linked gels can be described by the first derivation of W with respect to γ:
τ = τ1+ τ2 = ∂W
∂γ = ∂W1
∂γ +∂W2
∂γ The derived expressions for τ based on W1 and W2 are:
τ1 = G1(γ − γS) and τ2 = G2(γ + γi− γS)
For a network at ease (γ = 0) the two stress contributions from the structures are balanced, meaning τ1+ τ2 = 0. When substituting the derived formulas an equation for the residual strain can be acquired (equation 3).
γS = G2
G1+ G2γi (3)
When evaluating the shear modulus G for the different doubly cross-linked gels the apparent shear modulus (Gapp) can also be used. Instead of using the experimentally acquired value for the shear modulus an approximated value calculated by the measured shear stress during the applied shear strain can be used:
Gapp = τ γi
The apparent shear modulus is a way to help account for stress relaxation, nonlinear behavior at larger strains and nonlinear behavior at higher temperatures for certain polymer gels [13].
2.5 Controlled parameters
The parameters for this thesis were based on earlier experiments of the elasticity for regular Tetra-PEG gel [7]. The five parameters and their values were chosen to give comparable data to earlier findings. The five parameters that were controlled during the thesis were temperature, molecular weight of the precursors, connectivity, concentration of the precursors and the applied shear strain.
2.5.1 Temperature, T
The temperature interval of 5-25◦C was chosen based on earlier research [7]. If the temperature goes below 0◦C the water in the gel would freeze and if the temperature considerably exceeded 25◦C the gel might shrink due to large elasticity of the network.
The temperature was controlled using a thermostat in connection to the instruments during measurements. The model was an EYELA NCB-1200.
2.5.2 Molecular weight, M
The molecular weight of Tetra-PEG-MA and Tetra-PEG-SH used during the thesis was 20kg/mol (M = Mw = 20kg/mol). This is controlled by the length of the repeating unit of the precursor, which in return controls the size and therefore weight. All precursors used were provided by SINOPEG (XIAMEN SINOPEG BIOTECH CO., LTD.). The company provided Tetra-PEG-MA and Tetra-PEG-SH in powder format.
2.5.3 Connectivity, p
The connectivity of the polymer refers to the probability of connection between functional groups in the network structure [7]. In other words this means how much the two precursors are connected (figure 8). This can be controlled by the ratio of the two precursors to each other. If the ratio of the two is equal, meaning 0.5:0.5, a gel with a theoretical 100% connectivity is obtained. However if the ratio is changed the resulting connectivity will decrease. The two precursors A and B are mutually reactive and only react with each other. This means that only A-B cross-linking can occur, A-A or B-B cross-linking will not occur. So whichever precursor has the lower ratio will become the limiting precursor. For the rheological measurements only a mixing ratio of 0.5:0.5 was used.
Figure 8: Illustration of the connectivity of Tetra-PEG gels
2.5.4 Concentration, c
The concentration studied during this thesis was 60g/L. The needed amount of the precursors was calculated to acquire the desired concentration (equation 4).
m = Vtot× c × ratio (4)
Where m [g] is the mass of the precursor, Vtot [L] is the total volume of diluted precursor desired, c [g/L] is the desired concentration and the ratio is the ratio of the precursor.
Then to achieve the desired concentration with the actual measured amount of precursor it was diluted with CPB (equation 5).
VCPB = mreal
c (5)
Where VCPB [L] is the volume of CPB, mreal[g] is the actual weighed amount of precursor and c [g/L] is the concentration desired.
2.5.5 Applied strain, γi
The doubly cross-linked gel was fabricated by shear strain and the strain dependence of the shear modulus was studied. The applied strain is dimensionless, but will be presented as percentage during this thesis. For regular Tetra-PEG gel the negative energy elasticity was studied and showed a linear elasticity for τ , τE and τS for a wide range of strain up to γ = 140%. To study how the doubly cross-linked gel behaves fabricated by a variety of strains the applied strains of 25%, 50%, 100%, 200%, 400% and 800% was studied.
3 Experimental
The rheometers used were connected to a thermostat and a computer. The program RheoPlus was used to collect the data for all measurements.
3.1 Equipment and material
3.1.1 Equipment
Anton Paar Physica Modular Compact Rheometer 301 Anton Paar Modular Compact Rheometer 302
EYELA NCB-1200 thermostat Thinky AR-100 conditioning mixer
SHINRYO electron beam sterilized 10ml containers with lids (article number: 17823) 3.1.2 Material
Tetra-PEG-MA (M = 20kg/mol) Tetra-PEG-SH (M = 20kg/mol)
CPB (pH=3.8, concentration = 200mM)
3.2 Procedure
The rheometers were first calibrated according to the makers instruction before every measurement. For each measurement a total of 4.8ml Tetra-PEG gel solution was needed.
The needed amount of precursors were weighed and placed in separate containers. The calculated amount of room-temperature CPB was added to acquire the right concentration.
Then the container were respectively mixed by program 1 on the conditioning mixer (table 1). Then 2.5ml of the Tetra-PEG-MA solution was mixed with 2.5ml of the Tetra-PEG- SH solution by program 2 (table 1). A volume of 4.8ml of the mixed solution was then pipetted directly into the cylinder of the rheometer and the measurement was started.
After the start of the measurement a piece of wet tissue was placed around the cylinder edges away from the sample and the instrument was covered with a silicone cover.
3.3 Measurement program
A measuring program was created for the measuring of applied strains of 25%, 50%, 100%, 200%, 400% and 800% (table 2). The gelation of the 1st network equaled that of the gelation of regular Tetra-PEG gel. Then shear strain, γi, was introduced to create the second network structure. After the applied shear strain was removed the doubly cross-linked gel had been fabricated and any residual strain, γS, in the structure was measured. Finally the temperature dependence of the doubly cross-linked gel was measured for temperatures T of 5-25◦C while the angular frequency ω [rad/s] was sweeped from high to low frequency. The interval for the angular frequency sweep was 62.8-0.628 rad/s, equalling 10Hz to 0Hz. The temperature was measured from 25◦C to 5◦C and then back to 25 ◦C.
Table 2: Measurement program for the rheological measurements
Measurement Variable Constant Time [s] Data collected
Gelation of 1st network 25◦C, 1Hz 3200 G’, G” [pa]
Application of γi γi [%] 25◦C 46 000
Residual strain γS [%] 25◦C 900 γS [%]
T-dependence ω [rad/s], T[◦C] 13 400 G’, G” [pa], τ [Pa]
4 Results and discussion
4.1 Conclusions from compression testing on Tetra-PEG gel
During the first part of the thesis the negative energy elasticity was investigated for regular Tetra-PEG gel using compression testing. The aim was to confirm earlier findings with another measuring method than rheological measurements and then to investigate other polymer gels. However due to the global governing situation these measurements could not be completed as planned. Therefore the acquired results and discussion are presented in the appendices (Appendix A) and another project was commenced and the direction of the master thesis was adjusted.
The conclusions based on the compression testing of Tetra-PEG gels with M = 20kg/mol, c = 60g/L and different p was that the results seemed to indicate negative energy elasticity as found by earlier research [7]. However there were significant errors with the obtained results and therefore measurements to identify the problem became the main focus.
Several problems were noted surrounding the measurement including a non-leveled sample holder and shrinking of the samples during measurements. These factors were planned to be investigated to optimize the measuring method, but could not be completed as the new sample holder did not arrive in time. While waiting for the new sample holder other tests than compression tests were done. The main focus was shrinking tests of the gel in different oils as the sample was immersed in oil during testing. In the end pure silicone oil seemed to shrink the sample the least out of tested emulsions based on silicone oil, liquid paraffin and vegetable cooking oil. How much the shrinking affected the results was left uninvestigated. As inconclusive results were acquired for the Young’s modulus a tensile test of Tetra-PEG gel was also conducted. Based on all of the results from the compression tests and tensile test it was noted how easily the value for Young’s modulus is affected by different factors such as measuring method and calibration method. This is a known fact, but one that might not be advertised enough. As the value for Young’s modulus can differ significantly just based on the measuring method, for example tensile test or compression test, it is important to notice how the value was acquired before using literary values to compare with obtained results. After the arrival of the new sample holder it was planned to finish optimizing the compression testing method and then move onto other polymer gels to investigate the universality of the negative energy elasticity.
4.2 Rheological measurements on doubly cross-linked gel
The doubly cross-linked gels were based on Tetra-PEG gel by Tetra-PEG-MA and Tetra- PEG-SH with M = 20kg/mol, c = 60g/L, mixing ratio 0.5:0.5 and CPB (pH = 3.8, c = 200mM) as diluent. The measurements had four main parts including the gelation of the
regular Tetra-PEG gel, application of a shear strain, measuring of the residual strain and the temperature dependence of the shear modulus (table 2). These measurements were repeated for applied shear strains of 0%, 25%, 50%, 100%, 200%, 400% and 800%.
4.2.1 Shear modulus for the first network structure
The results from the gelation of the regular Tetra-PEG gel, meaning before applying any shear strain, showed that there were some fluctuations for the value of G1, meaning the shear modulus for the first network structure (figure 9, 10, 11 and 12).
Figure 9: G’ and G” plotted against time for the measurement with applied shear strain of 0%
Figure 10: G’ and G” plotted against time for the measurement with applied shear strain of A) 25% and B) 50%
Figure 11: G’ and G” plotted against time for the measurement with applied shear strain of A) 100% and B) 200%
Figure 12: G’ and G” plotted against time for the measurement with applied shear strain of A) 400% and B) 800%
The value for G1 was approximated by the last saved value for G’ for every measurement of the first network structure as G” was approximately zero for all measurements. This value was expected to be the same for all measurements as the regular Tetra-PEG gel was the same for all doubly cross-linked gels. In the figures the corresponding value for G1app, the apparent shear modulus, for the first network structure is also presented. This value was calculated using the obtained shear stress, τ , from the shear modulus of the doubly cross-linked gel when applying the shear strain, γi, later. The stress values are available in the appendices (Appendix B).
The values for G1 that was expected to be approximately the same showed a wide spread (table 3). Comparing the values it became clear that there was some larger error for measurement 1 and probably also measurement 6. With the exception of the value from the measurement 1 G1 were within 7% of each other. Further excluding the value from measurement 6 G1 was within 3% of each other. Now a fluctuation of 3% was deemed reasonable due to the fact that the connectivity of the samples fluctuates some when making the gel and the fact that two different rheometers were used. When the two rheometers were compared they gave results within ±2% of each other. However the
value of 3650 Pa (figure 9) and 4520 (figure 12 A) Pa were deemed unusually low and indicated some other error with those measurements.
Table 3: G1 values for all measurements
Measurement 1 2 3 4 5 6 7
Later applied γi [%] 0% 25% 50% 100% 200% 400% 800%
G0 ≈ G1 [Pa] 3650 4770 4810 4860 4820 4520 4720
Looking at values from measurement 6 and 7 it can also be seen that these are lower than the acquired values for measurement 2-5. This might indicate the results were affected by nonlinear effects. It was for example theorized that larger applied strain could cause a weak damping of the shear modulus, which could help explain the lower acquired values.
4.2.2 Stress dependence of the applied shear strain
When calculating the apparent shear modulus for the first network the values for the shear stress, τ , was used. The shear stress was collected from the measurement where the shear strain was applied. The mean value for the shear stress was based on 2300 data points for each sample. The stress values in general only showed slight fluctuations and the same shear stress was shown for an overwhelmingly large part of the data points. The results from the collected values from separate measurements showed some interesting results.
According to acquired stress data the stress increased linearly up to γi = 100% (blue circles figure 13). But after γi = 100% it decreases. A decrease in the shear stress was expected due to nonlinear effects, but not as early as γi= 100%. Then the results showed that the stress suddenly increased again for γi = 800%, which is also a surprising result.
To be able to confirm the results another stress dependence test was done on the same sample instead. The stress in this case was a mean of five measurements at the beginning of the gelation of the second network structure. The acquired results showed a constant linear increase of the stress until γi= 200% and then a decrease for γi= 400% and 800%
(red squares figure 13). These results coincided better with the expected results of an increase in stress and decrease at larger strains due to nonlinear effect and damping. The lower strain region stress is very similar for both types of measurements it is only in the larger strain region the results differ significantly.
Figure 13: The stress dependence of the applied shear strain
The results would indicate that the first acquired results might have been affected by some error or nonlinear effects. Earlier reports on the nonlinear behavior indicate that the shear stress should increase linearly until the larger strain region where the stress should decrease. For earlier studies on the regular Tetra-PEG gel the decrease was seen at γi = 200%, which coincide with the results from the same sample. However it should be noted that the measurement settings were different for the measurements. For the separate measurements the time between each measurement was 20 seconds. For the same sample it was only 1 second. This could be problematic for larger strain regions as the viscoelastic properties are affected. The polymer chains do not for example have time to relax with a shorter interval time. To properly confirm the stress dependence a stress relaxation test should be conducted. Further it should be noted that the separate measurements where conducted during the full gelation process, but the measurements for the same sample were completed during the beginning of this process.
4.2.3 Study of the residual strain
After the removal of the different applied shear strains the gel was left with two network structures. The experimentally obtained value for the residual strain, γS, did not show a linear relation when plotted against applied shear strain, γi (black circles figure 14).
For the plotted experimental data one can see a distinctive difference for γi lower and higher than 100%. The lower strain region (γi < 100%) seems to show linearity and the higher strain region (γi > 100%) seems to do the same. But the whole of the shear strain interval cannot be well-described by a single linear regression. This would indicate that the residual strain for the lower strain region and higher strain region has to be considered separately.
Figure 14: Experimental data and theoretical estimate for the residual strain and two different models based on exponential fitting of the values for G1app
The calculated theoretical values for the residual strain, using equation 3, overlap well for the lower strain region (table 4)(pink line figure 14). This result would indicate that the lower shear strain region can be well-described by the two-network theory and classic rubber theory. However the theoretical estimation for the higher strain region do not overlap well with the experimental data. To be able to derive a model that can describe all of the obtained data the use of exponential fitting was needed instead of linear fitting.
The exponential fitting was applied to G1app and the applied shear strain. The obtained exponential expression was then used to approximate values for G1 (Appendix B). This value could then be used to calculate G2 using the experimentally obtained values for G (G = G1 + G2). Then the residual strain was calculated for all applied shear strains (equation 3). The fitting seemed to fit quite well with the experimental data (red line figure 14). However earlier data had indicated some larger error with the measurement for γi = 400%. Hence the exponential fitting was re-done without this value and the fitting seemed to describe the experimental γS even better (blue line figure 14). This result would further support the fact that there might have been some problems with this particular measurement. Regardless it does seem that the experimentally obtained residual strain can be quite well-described by an exponential fitting of G1app (table 4).
Table 4: Comparing values from the experimental data, theoretical approximation and models for the residual strain based on exponential fitting of G1app
Applied shear strain γi [%] 25 50 100 200 400 800
Experimental γS [%] 10.5 21.3 41.9 145 351 754
Theoretical γS [%] 10.7 21.6 41.9 84.6 178.7 357.1 Fitting based on all γi [%] 10.4 23.5 54.0 133.4 328.1 761.6 Fitting omitting γi=400% [%] 10.4 23.7 57.9 142.7 356.7 757.5 This would indicate that the experimentally acquired residual strains for doubly cross- linked gel fabricated by shear strain can be described by:
γS = C
A × eB ×γ i+ C × γi
Where γi is the applied shear strain, γS is the residual strain, C is the apparent shear modulus for the second network structure. A and B are constants for the exponential fitting of G1app according to y = A × eb×γ i.
4.2.4 Temperature dependence of the shear modulus
After fabricating the doubly cross-linked gels the temperature dependence and stability of the shear modulus was investigated. The temperature dependence of the shear modulus was studied for the interval of 5-25◦C. The temperature values of 30-40◦C were studied as well, but excluded when studying the elasticity as they showed nonlinearity effects. Earlier research on Tetra-PEG gel expected this behavior due to large elasticity of the polymer network when the temperature exceedT > 298K [7]. Both values for G’ and G” were acquired, however for all of the measurements the G” was approximately 0. This would indicate that the shear modulus of the doubly cross-linked gel could be approximated by G’. Hence only G’ was plotted in the figures below. A total of 20 data points were acquired for every measured temperature and out of those a significant number showed the same G’-value. The used G’-value was approximated by the mean of all data points for every temperature.
Based on earlier research it was expected that the temperature dependence of the doubly cross-linked gel would mimic the results of the regular Tetra-PEG gel they were based on [7]. According to earlier findings T0, meaning the x-axis intercept in a plot of the shear modulus as a function of temperature, was independent of the connectivity p and only dependent on M and c. The T0 for Tetra-PEG gel with M = 20kg/mol, c = 60g/L and a mixing ratio of 0.5:0.5 was found to be 131K from previous research [7]. For the fabricated doubly cross-linked gels the only parameter out of these changed was the connectivity p due to the second network structure. Hence according to earlier findings T0 should be the same regardless of the applied shear strain. However the result for the regular Tetra-PEG gel which all of the doubly cross-linked gels were based on (γi= 0%) showed a temperature dependence that significantly deviated from the expected value of T0 = 131K (figure 15 A). The acquired T0 was 137.8K. This value was much higher than expected. To try and see how the extrapolation was affected by the expected T0 the data was refitted with the addition of the data point (131,0) (figure 15 B).
Figure 15: Temperature dependence of G A) from experimental data and B) from experimental data + the added point (131, 0)
Looking at the results it becomes clear that the x-axis intercept T0 is very sensitive to fluctuations of the G’-value. When forcing the linear regression through the expected intercept of (131,0) R2 is actually improved. Based on this since we are extrapolating the line until the x-axis intercept for all measurements some fluctuations was expected for the other data as well. It should be noted also that the actual shear modulus and shear stress do not follow the extrapolated lines at lower temperatures than the measured temperatures. Furthermore it doesn’t actually vanish at T0. The extrapolated G-T relation was studied based on earlier research were the aim was to calculate the energy and entropy contributions of the elasticity [7].
The acquired T0 values for the lower and higher strain regions were closer to the expected value of 131K (figure 16 and 17 A). Based on the acquired results it would seems like the value of T0 = 137.8K for γi = 0% was unusually high. This could support the earlier impression that there might have been some larger error affecting the result during this particular measurement.
Figure 16: Temperature dependence of G for the doubly cross-linked gel for applied shear strain of A) 25% and B) 50%
Figure 17: Temperature dependence of G for the doubly cross-linked gel for applied shear strain of A) 100% and B) 200%
