Bending rigidities of a patchy particle gel

(1)

Bending rigidities of a patchy particle gel

Kasper van Tulder

September 9, 2020

Study Physics & Astronomy (UvA/VU) Course Bachelor Project

Research group Soft Condensed Matter Physics, Van der Waals-Zeeman Institute Course Bachelor Project

Supervisor prof. dr. Peter Schall Second examiner prof. dr. Peter Bolhuis Daily supervisor Piet Swinkels

(2)

Abstract

Patchy particles are colloidal particles with a limited valency. These particles have directional patches which allows the particle to interact with other particles through their patches. This happens when the particles are dispersed into a binary liquid close to the critical temperature. When two particles are close to each other, the thermal fluctuations will be confined by the boundary conditions of the surfaces of the particles. This results in an effective bonding force between the particles, known as the critical Casimir force. This force is temperature dependent: the magnitude of the force increases as the temperature approaches the critical temperature.

In this study, we researched the effects of incorporating dipatch chains into a larger network. Dipatch particles in a 25% 2,6-lutidine solution were injected into a glass capillary. The sample was heated to start a self-assembly process and dipatch chains were formed. Another sample was prepared consisting of a mix of dipatch and tetrapatch particles. The tetrapatch particles function as branch points to connect the dipatch chains. This results in a semi-solid percolated gel.

Both samples were analysed at 33.55°, 33.6° and 33.65° Celsius. Microscopic images were made of both samples and video analysis is used to determine the position of each particle. First, we calculated the persistence length of the dipatch chains for each temperature. Secondly, we compared the variance of the angle in the gelled network with the variance in the dipatch chains.

A positive trend was found for the dipatch chains: the persistence length increases with higher temperatures. This corresponds with our expectation as the critical Casimir force increases when the temperature approaches the critical temperature resulting in a stiffer bond. In contrary to what one might expect, the chain strands in a patchy gel are not stiffer then the unconstrained dipatch chain. The stiffness of chains in the network is identical to the stiffness of a free chain, meaning the network does not impose stress on the chains.

(3)

Populaire samenvatting

In dit onderzoek wordt gebruikt gemaakt van patchy deeltjes. Deze deeltjes zijn erg klein, maar wel groter dan atomen. Ze zijn nog wel klein genoeg om bepaalde eigenschappen met atomen te delen, maar ze zijn ook groot genoeg om te bestuderen met een microscoop. Dat maakt deze deeltjes erg interessant: we kunnen ze onder andere gebruiken om atomen beter te begrijpen. Atomen kunnen bindingen aangaan met andere atomen, waardoor moleculen gevormd worden. Het hangt af van het soort atoom hoeveel bindingen dit zijn. Zuurstof kan zich bijvoorbeeld binden aan 2 andere atomen, terwijl koolstof dit met 4 andere atomen kan. Net als atomen kunnen onze deeltjes ook deze bindingen aangaan. Hoeveel bindingen dit zijn hangt af van hoeveelheid ‘patches’. Er zijn deeltjes met twee van deze patches en deeltjes met vier.

In dit onderzoek heb ik onder andere gekeken naar lange kettingen van deeltjes met twee patches. Ook heb ik gekeken naar een zogenoemde gel: als we een mix vormen van deeltjes met 2 patches en af en toe een deeltje met 4 patches, resulteert dit in allemaal kettingen die verbonden zijn met elkaar.

Eerst heb ik gekeken naar de stijfheid van de kettingen. Dit heb ik gedaan door te kijken hoeveel de deeltjes onderling bewegen. Omdat de deeltjes elkaar harder aantrekken bij hogere temperaturen, verwachten we dat de kettingen en de gel stijfer zullen zijn als de temperatuur hoger is. Dit hebben wij ook terug gevonden in onze resultaten.

Vervolgens hebben we ook gekeken naar de stijfheid van de gel. Doordat de kettingen aan elkaar geplakt zijn, verwachten we dat de kettingen minder makkelijk kunnen bewegen. Hi-erdoor verwachten we dat de gel stijfer zal zijn dan losse kettingen. Dit hebben we alleen niet terug gezien in de resultaten, de kettingen in de gel blijken even stijf te zijn als de losse kettingen.

(4)

1 Introduction

Self assembly, the process of spontaneous formation of ordered structures due to local inter-actions, can be seen everywhere: atomic self-assembly, the formation of crystals and protein folding. Three ingredients are needed for self-assembly: a building block, a local interaction that causes the bond and a source of movement [1]. These three ingredients are not restricted to self-asembly systems, but also hold up for most built structures: for example, a man-made brick wall can also be divided into those categories. The brick would be the building block, the cement is the local interaction that holds the bricks together and the source of move-ment would be the construction worker. An interesting choice for the building block in our self-assembly system would be a colloid, a particle with a size in the order of a few hundred nanometers, up to the micrometer scale. Because of this larger scale compared to atoms, col-loids are completely classical and have no unusual optic or magnetic properties [2]. Although a single colloidal particle may not show interesting properties, many particles together show intriguing collective properties. Combined with the fact that they are still big enough to study them with brightfield or confocal microscopy, but also small enough to still experience Brownian motion, they are ideal objects to study self-assembly: the Brownian motion ensures that no external motion source is needed, while the bigger size allows us to easily study them in real time.

Aside from the movement source, a bonding force is also needed before the particles can self-assemble. This can be achieved by dispersing the particles in a binary liquid close to the critical temperature. An effective attractive force arises between the particles due to the confinement of the solvent, known as the critical Casimir force. This force is the thermo-dynamic equivalent of the quantum Casimir interaction [3]. The temperature dependence of the critical Casimir force allows us to easily control the magnitude of the force: because there is no attraction at room temperature with certain solvents, like 2,6-lutidine [4], it is possible to heat up the sample to a few degrees below the critical temperature and start a self-assembly process. If the solvent is cooled down again, the bonding force disappears and the assembly is reversed. Using the critical Casimir force, we can easily study the formation process of complex structures under different conditions. By altering the temperature, and thus changing the distance to the solvent critical point, we can study this formation process with variance in the bonding force.

Self-assembled colloids can have many different structures, and even more can be unlocked by changing the shape or surface of our building block [2]. By designing particles to be anisotropic, we unlock even more exciting self-assembled structures eventually leading to newly designed materials [5], like the kagome lattice[6]. Because crystallization and phase transitions of colloids rely on the same underlying physics as that of atoms, we can use colloidal particles as an analogue model to study crystallization and phase transitions of atoms [2, 7]. Due to recent advances in colloidal synthesis, colloidal particles can be made with more control over shape and interaction [8]. For instance, Gong et al. have created a new kind of colloidal particles which have a fixed maximum amount of valency.

(6)

Figure 1: A schematic phase diagram of a binary liquid. On the x-axis is the concentration and on the y-axis is the temperature. The red area shows the temperatures Tcx where the

binary liquid separates into two components. The image is taken from [1].

This limited valency is achieved by a technique called colloidal fusion. The four spheres are melted, thus extruding the liquid core to the surface. This results in four patches at the surface of the particle. A similar technique can be used with three outer spheres to create a dipatch particle. These techniques are visualised in figure 2. Simulations have shown that the directional encoded location of the patches, combined with the limited valency, allows the formation of complex structures like a diamond structure[9]. Recent unpublished work has also shown the formation of penta-rings[10].

Figure 2: The synthesis of patchy particles with respectively one, four or eight patches. The synthesis is done by colloidal fusion: a certain a amount of spheres are melted onto a softer core, resulting in a extrusion to the surface. The image is taken from [8], which also includes a more detailed description of the synthesis.

(7)

The critical Casimir force also causes an attraction between normal, non-patchy colloids. If this inter-particle attraction becomes bigger than the thermal energy kBT then the colloids

will cluster. If this happens slowly and carefully, the particle will form a crystal. However, when we heat the sample quickly such that the interaction is much bigger then kBT , the

sample may not reach equilibrium, and instead form a gel. Such a gel is in a non-equilibrium state: the particles slowly restructure trying to complete the phase separation [11]. It is also possible to create an equilibrium gel with patchy particles. This can be done by creating a mixture between dipatch and tetra-patch particles. The dipatch polymers can form connec-tions with other polymers through the tetra-patch branch points and form a semi-solid gel. In contrary to the non-patchy gel, this gel is in equilibrium meaning it is a thermodynamically stable state. stability[11]. The thermodynamic stability is interesting, because it results in a non-aging gel. This can be useful for consumer products, where aging results in a shorter shelf life.

In this thesis, we study the stiffness of such a patchy gel. Because the gel mainly consists of dipatch polymers we also investigate the bending rigidities of free dipatch polymers. How do these polymers compare with the network gel? It is expected that the network gel will be stiffer than the unconstrained polymers, as the branch points add an extra constraint to the movement of the polymers.

2 Worm-like chain model

Many models exist to describe polymers, like the simplified ideal chain model or more com-plicated models like the worm-like chain model [12]. To describe our dipatch chains we are in need for a model to describe the stiffness. A quite simple model is the freely jointed chain model[13]: This model assumes that the chain consists of multiple parts of fixed length, just like our dipatch particles. The model also assumes that the parts are connected with a free joint: the bonds have no angle constraint. This is not the case in the dipatch chain where the bond angles are close to 180° degrees and can not move freely. A better model is the freely rotating chain model [13]: this model assumes both a fixed length and a fixed bond angle, but the particles can still rotate around the bond.

An interesting choice for our dipatch chain would be the Worm-like chain model. This model is the limiting case of the bond length l→0 and the bond angle θ →180°. Because the particle bond angle always approximates 180 degrees and a single patchy particle is way smaller then the whole chain, we can use the worm like chain model. Instead of a discontinous chain, we now describe our chain as a semi-flexible continuous rod [14]. The flexibility of this rod can be expressed by the persistence length: the length over which the direction of the tangent no longer correlates [15]. The persistence length for a 2 dimensional worm-like chain can be expressed by h ~R · ~Ri = Z L0 0 ds Z L0 0 ds0e−|s−s0|/2P (1)

where P is the persistence length, L0 is the total arc length and < R2 > is the mean squared

(8)

hR2i = 4P L0 1 −2P L0 eL0/2P (2)

A taylor expansion is made of the exponential term up to the fourth term:

hR2_{i = 4P L} 0 1 8 L0 2P − 1 6 L0 2P 2! (3) hR2i = L 2 0 2 − 1 6 L3₀ P (4)

With some basic algebra it is possible to rewrite this as: P = 1

6

L2₀

L2₀− < R2 _> (5)

With this equation we can express the persistence length when we know the location of all the particles in the chain. Note that this formula is only valid for L0 << P because of our

Taylor expansion. It may also be useful to express the persistence length in terms of the angle distribution of the chain bonds. This can be done in the following way

P = d

σ2 (6)

where P is the persistence length, d is our diameter of the particle and σ is the standard deviation of the angle.

(9)

3 Method

In this thesis, we use two different samples to measure the effects of incorporating a colloidal polymer into a larger network. First, a dipatch polymer was imaged at three different tem-peratures. Secondly, a network gel was imaged under the same conditions and temtem-peratures. The dipatch polymers and the network gel data were both gathered by S. Stuij. The worm-like chain model was used to determine the persistence length of the dipatch polymer. Using this persistence length, we can judge whether strands in the gel have higher, lower, or equal stiffness as a free chain.

3.1 The dipatch chains

The dipatch particles were prepared in a 25% 2,6-lutidine and 75% milliQ water mixture. 1 mmol MgSO4 is added to screen the charge of the dipatch particles. This mixture is injected

into a glass capillary and left at room temperature until all particles are sedimented to the bottom of the sample.

The samples are heated to a temperature of 33.55°, 33.60° and 33.65° Celsius using a carefully controlled waterbath. Brightfield images were captured at 10 frames per second for at least 90 seconds.

Trackpy [?], an open source Python library, is used to analyse each gathered video. With help of this library, we determine the position of each particle for every frame of the acquired images. Trackpy’s algorithm achieves this by searching for bright features in the images. To prevent false positives, a band-pass filter is applied to remove most of the small wavelength noise. The remaining peaks in brightness due to noise are filtered out by requiring a minimum of Trackpy’s mass quantity. This mass-like quantity is calculated by integrating the total brightness of the particle. Because there should only be movement due to drift and Brownian motion, we can use the implemented Crocker-Grier algorithm to link every particle over time[16].

A clustering algorithm is used to calculate which particles are bonded. Two particles are considered bonded when they are roughly 1.2 microns apart (the diameter of our particle). For our analysis, we only select polymers consisting of more then 10 particles. We then calculate the persistence length of each polymer chain using equation 5. This procedure is visualised in figure 3

The persistence length is also calculated in a alternative way. We calculate the angle for every particle in a chain with its neighbour. By acquiring enough data points, it is possible to express the persistence length with equation 6.

(10)

Figure 3: An example of a analysed dipatch polymer. The green circles are the detected particles by trackpy. There are still some false positives, but they are filtered out by the cluster algorithm as described in section 3.1. Every image is visually inspected for false positives to see if they form any problems. The length of the colored line represents the calculated arc length. The color represents the angle with it neighbours. The dashed black line represents the end-to-end distance.

(11)

3.2 The gelled network

Using the same particles, we now assemble a network of free chains. By introducing tetrapatch particles as branching points, and increasing the particle concentration, a percolated network forms. The network gel consists of a mixture of dipatch and tetrapatch particles with a number ratio of 5 : 1 dispersed in 25% 2,6-lutidine and 75% milliQ water. Just like the free dipatch polymers, 1 mmol MgSO4 is added to screen the charge of the particles. The

mixture is injected into a glass capillary and left at room temperature until all particles have sedimented to the bottom of the sample. The sample is put on top on a waterbath and heated to 33.5° , 33.6° and 33.65° Celsius.

Just like the free chain case, we use Trackpy to determine every particle’s position. Again, a bandpass filter is applied and false positives are filtered out by requiring a minimum bright-ness. However, because confocal microscopy is used, the bright spots in our images are now the fluorescently labeled patches instead of the particles itself. Some extra steps are required to reconstruct the original particle position. First, we need to find the other patch which is part of the same particle. This is done by enforcing a distance range between two patches before it is considered to be a particle. Secondly, we need to reconstruct the original position of the particle, which is done by taking the average position of the two patches. This proce-dure is visualised in figure 4. The next step is to track the particles over time and calculate the angle with its neighbours for each frame. As a final step, the standard deviation of this angle is calculated and compared with the dipatch chain at the corresponding temperature.

Figure 4: An representative example of a analysed network dataset. This is a single strand taken from the full network matrix. The dataset is recorded at a temperature of 33.55° Celsius. Every red circle is a recognized patch. The white circles are the interpolated locations of the particles.

(12)

4 Results and discussion

4.1 The dipatch chains

We have analysed several different datasets of the dipatch chains at different temperatures. We determined the position of every particle for every frame in each dataset. This showed that the chain wiggles around during the measurement, although it is cleary quite rigid. The particles stay in the chain for the duration of the measurement, meaning that the chains are stable. An example of a frame of an analysed dataset is shown in figure 3. The persistence length was calculated using two different methods.

Firstly, the persistence length was calculated by comparing the total arc length with the end-to-end distance. L0 and R were calculated by summing over all particles in the chain in the

following way:

L0 = ΣN −1i=0 p|xi− xi+1|2+ |yi− yi+1|2 (7)

R =p|x0− xN|2+ |y0− yN|2 (8)

where L0 is the arc length, xn, yn is the nth particle position in a polymer and R is the

end-to-end distance. With L0 and R known, the persistence length was calculated using equation

5.

Secondly, we make use of the individual variation in angles. For every particle in the chain, we calculate the angle with its neighbours. A Gaussian distribution is fitted to the angle distribution to determine the standard deviation. The persistence length is calculated from the standard deviation using equation 6.

The persistence lengths obtained with both methods are shown in figure 6. The persistence length clearly increases with increasing temperature. This can be explained by the temper-ature dependence of the critical Casimir force: as the tempertemper-ature gets closer to the critical temperature, the solvent correlation length also increases. This results in a higher critical Casimir force and thus in a stronger, stiffer bond. This bigger force easily compensates for the thermal energy gain. Figure 6 also shows some difference in the values obtained by the two methods, especially for the 33.60° Celsius datasets. This can be explained by the fact that our arc-length vs end-to-end method is more prone to errors: the requirement that the chain should at least consist of 10 particles reduced our sample size significantly, as we don’t meet the requirement anymore if one of the middle particles gets out of focus.

The two results of 33.6° Celsius were combined in one histogram to get a better fit of the stan-dard deviation, the same was done for the 33.65° Celsius sets resulting in values of respectively

(13)

Figure 5: a (left): An typical example of a angle distribution for a dipatch polymer. The shown distribution of of dataset 3 with a temperature of 33.65° Celsius. A gaussian distribution is fitted and shown in orange.

b (right): All the gaussian fits of the angle distribution for every temperature.

Figure 6: The found values for the persistence length. The blue datapoints are the values gathered by comparing the end-to-end distance with the total arc length, as described in section 3.1. The orange datapoints are calculated by looking at the angle distribution, as described in section 3.1

(14)

4.2 The gelled network

When we increase the particle concentration and introduce tetrapatch particles to serve as branching points, we obtain a gel. An image of the created gel network is shown in figure 7. The gel consists of long strands of dipatch particles, which are connected to other strands through tetrapatch particles. Although the tetrapatch particles can form up to four bonds in principle, no particles were found with four bonds in practice. This can be explained by the sedimentation of the particles. We left the sample at room temperature until all particles were sedimented to the bottom of the sample. This should result in a 2D-like situation with a gravitational height close to the particle radius. The particles can still move up a little bit against gravity trough thermal motion, the gravitational height (h=kBT

Fg ) gives an

approximation how easy this is. When a tetrapatch particle forms four bonds, one bond is always pointing upwards, outside of the 2D plane. This makes it energetically unfavourable to have all four bonds filled. An example of that bonds can still move a bit up against gravity is shown in figure 7. There are some overlapping dipatch strands visible: the strands cross each others path, but are not connected with a tetrapatch particle.

Figure 7: The gelled network. The bright spots in the picture are the fluorescently labeled patches

We determine the stiffness of the gel strands by calculating the standard deviation of the variance of the bond angles: First, the position of every particle is determined for each frame. Secondly, a clustering algorithm is used to determine which particles are bonded. Finally, the variance in the angles for each particle is calculated and shown in a color coded plot of the network. A typical example of the local variance of the bond angle of the gelled network is shown in figure 8 (other datasets are available in appendix).

(15)

Figure 8: The analysed network for 33.55° Celsius. The values matching with deviation of the unconstrained polymer found in the section 4.1 are color coded white, values larger are color coded blue and more rigid particles are color coded red

(16)

Looking at figure 8, the first thing we observe is that branch points move less. This is not surprising considering that the branch points are more constrained in their movement. We also see that within a chain in the network, the stiffness is more-or-less equal throughout the chain. This matches with our expectations: because the particles are connected with each other, one particle can not move without moving its neighbours.

To compare the stiffness of the gelled network with the dipatch chain, a similar representation of the variance within the chain is made in figure 9. Similar results were found: the variance in standard deviation matches with its neighbours and no big variance was found.

The distributions of the standard deviation of the gelled network are plotted in figure 10. We see that the peaks match with our baseline values found in section 4.1. This is in contrast to our expectation that the peaks would be at lower values then our baseline, as the branch points in the gel add extra constraints in the movement of the dipatch chains. Instead we see that the peaks match: this suggests that the strands inside the gel network are not stiffer then the single dipatch chains.

(a) The standard deviation distribution for 33.55° Celius.

(b) The standard deviation distribution for 33.60° Celius.

(c) The standard deviation distribution for 33.65° Celius.

(17)

The distribution of the standard deviation from n samples taken from a set given by a normal distribution is given by a gamma-type distribution:

fN(s) = 2

N/2σ2(N −1₂ ) Γ(1₂(N − 1)) e

−N s2_/2σ2_sN −2

(9)

where fN is our distribution function, σ is the standard deviation of our gaussian distribution,

Γ(z) is the gamma function and N the degrees of freedom [17]. If all values vary around the same value with the same ’true’ standard deviation, this is the distribution you would expect to get. The expected distributions shapes for the gelled network at different temperatures are plotted in figure 11

Figure 11: The theoretic standard deviations distributions for the gelled network at 33.5°, 33.6° and 33.65° Celsius.

The shape of the theoretical expected distributions does not match with the experimentally found distributions: the experimental distributions do have a lot of higher standard devia-tions, while the expected distribution converge faster to zero. The high standard deviations are thought to originate from tracking errors, because false-positives have no fixed position, resulting in a high standard deviation.

(18)

5 Conclusion

In this work, we study the mechanical properties of an equilibrium patchy gel. We compare the stiffness of strands in the patchy gel with free colloidal polymers under identical conditions. To do this, we first look at the unconstrained dipatch chain. An expected trend was found: as the temperature becomes closer to the critical temperature, the Casimir force increases and the chain becomes stiffer. This bigger force compensates any thermal energy gain resulting in a bigger persistence length for a temperature closer to the critical temperature. Secondly, we look at the dipatch strands incorporated in the patchy network gel. In contrary to what one might expect, the polymer strands in a patchy gel are not stiffer than the unconstrained dipatch chain. The stiffness of chains in the polymer network is identical to the stiffness of a free chain, meaning the network does not impose stress on the chains.

Future research could focus on calculating the visco-elastic moduli by using passive micro-rheology, as this is a better approach to define stiffness. Another interesting choice could be looking at externally applied stress: how does the gel react under a external force? And how does the stress divide over the gel?

(19)

References

[1] S. G. Stui. Colloidal design: Building, bending and breaking. PhD thesis, Universiteit van Amsterdam, Amsterdam, 2020.

[2] V. N. Manoharan. Colloidal matter: Packing, geometry, and entropy. Science, 349(6251):1253751–1253751, August 2015.

[3] H.B.G. Casimir. On the Attraction Between Two Perfectly Conducting Plates.

[4] C. Hertlein, L. Helden, A. Gambassi, S. Dietrich, and C. Bechinger. Direct measurement of critical Casimir forces. Nature, 451(7175):172–175, January 2008.

[5] Stefano Sacanna, Mark Korpics, Kelvin Rodriguez, Laura Col´on-Mel´endez, Seung-Hyun Kim, David J. Pine, and Gi-Ra Yi. Shaping colloids for self-assembly. Nature Commu-nications, 4(1):1688, June 2013.

[6] Qian Chen, Sung Chul Bae, and Steve Granick. Directed self-assembly of a colloidal kagome lattice. Nature, 469(7330):381–384, January 2011.

[7] Anthony J. Kim, Paul L. Biancaniello, and John C. Crocker. Engineering DNA-Mediated Colloidal Crystallization. Langmuir, 22(5):1991–2001, February 2006.

[8] Zhe Gong, Theodore Hueckel, Gi-Ra Yi, and Stefano Sacanna. Patchy particles made by colloidal fusion. Nature, 550(7675):234–238, October 2017.

[9] Zhang, Aaron S. Keys, Ting Chen, and Sharon C. Glotzer. Self-Assembly of Patchy Particles into Diamond Structures through Molecular Mimicry. Langmuir, 21(25):11547– 11551, December 2005.

[10] Piet Swinkels and Simon Stuij. Revealing conformational dynamics of colloidal molecules. Technical report, Institute of Physics, University of Amsterdam, The Netherlands. [11] Francesco Sciortino and Emanuela Zaccarelli. Equilibrium gels of limited valence colloids.

Current Opinion in Colloid & Interface Science, 30:90–96, July 2017.

[12] R. A. Pethrick. The theory of polymer dynamics M. Doi and S. F. Edwards, Oxford University Press, Oxford, 1986. pp. xiii + 391, price£40.00. ISBN 0-19-85 1976-1. British Polymer Journal, 20(3):299–299, 1988.

[13] Gert R. Strobl. The Physics of Polymers. Springer Berlin Heidelberg, Berlin, Heidelberg, 1997.

[14] Andrew Marantan and L. Mahadevan. Mechanics and statistics of the worm-like chain. American Journal of Physics, 86(2):86–94, February 2018.

[15] Paul. J. Flory and M. Volkenstein. Statistical mechanics of chain molecules. Biopolymers, 8(5):699–700, November 1969.

[16] John C. Crocker and David G. Grier. Methods of Digital Video Microscopy for Colloidal Studies. Journal of Colloid and Interface Science, 179(1):298–310, April 1996.

(20)

6 Appendix

6.1 Trackpy’s accuracy

To estimate trackpys precision, a video is analysed of a particle stuck to the glass. Because it is known that there should be no movement, we can easily see the error caused by the analysis. The variance of the x and y axis are added together in figure 12a. The fitted gaussian distribution has a standard deviation of 0.40 pixels, which converts to 25 nm. Note that the accuracy is higher than the pixel density, trackpy achieves this by taking the average position weighted by brightness. The trackpy documentation states that a wrong mask size can influence this sub-pixel accuracy, creating a bias towards the natural numbers. The sub-pixel distribution is checked for every dataset and no bias was found. An example of a sub-pixel distribution is plotted in figure 12b.

(a) The variance in position of a particle stuck to the glass.

(b) The subpixel bias of dataset 8.

Figure 12: The distribution of the error in trackpy (left) and the sub-pixel bias (right)

6.2 End-to-end vs arc length accuracy

Beside the small sample size, there are other reasons to distrust these values. Sometimes, the particles at the end of the chain are not resting on the glass but pointing a bit upwards. Our persistence equation for the worm-like chain assumes that the chain is two dimensional, which is not the case if the end have a small z component. L0 is also a bit underestimated

for these end points, as the z component cannot be seen from the top. This can effect our persistence length to be on the high side.

The angle distribution method as discussed in section 3.2 doesn’t have the previously discussed problem. If one particle is out of focus and not recognized, it is still possible to calculate the

(21)

6.3 Attached images

Figure 13: The analysed network for 33.55° Celsius. The values matching with deviation of the unconstrained polymer found in the section 6 are color coded white, values larger are color coded blue and more rigid particles are color coded red

(22)

(23)

Bending rigidities of a patchy particle gel