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Colloidal particles or shortly colloids are objects with the dimensions in the range roughly between 100 nanometers to 1 micron. When these particles are dispersed in a liquid medium we get a colloidal suspension. The examples of suspensions from everyday life include milk, paint, blood, etc. One of the key properties of these systems is their stability. Particles can be either stable and well dispersed in the medium or they can stick together to form aggregates. Both processes are important. For example in the paint we want the particles to be stable and not to aggregate and sediment to the bottom of the paint container. On the other hand in the water waste treatment, we want that the fine dirt particles aggregate and sediment in order to separate them from the clean water.

Stability of the suspensions can be measured experimentally by Light Scattering techniques. During the aggregation process the change of the scattering intensity is monitored (Static Light Scattering) and/or the change of the apparent hydrodynamic radius (Dynamic Light Scattering). From this changes the aggregation rate can be calculated and hence the stability ratio [1]. By combining the two techniques into Simultaneous Static and Dynamic Light Scattering (SSDLS) one can measure also the absolute aggregation rate constant [1].

Figure 1. Animation of aggregation of colloidal particles (left) and schematic stability plot of the colloidal suspensions versus the coagulant concentration (right, Source: www.wikipedia.org).

Colloidal suspensions can be also treated theoretically. The classical DLVO theory was developed in 1940s and has successfully explained the stability of the charge particles in the presence of the electrolytes. The basis for this theory is the competition of the attractive van der Waals forces and repulsive electrostatic forces, which compete to either stabilize or de-stabilize the suspension [2, 3]. If the sum of the two contributions is attractive the particles will aggregate, while the repulsive sum leads to the stable suspension.

Figure 2. When the sum of the electrostatic and van der Waals forces is repulsive the suspension is stable (upper figure), whereas the aggregation in the suspension occurs when the sum is attractive (bottom).

The electrostatic part of the DLVO forces is usually treated within the Poisson-Boltzmann theory. The theoretical results agree well with the experimentally measured interactions with AFM [4, 5]. Also the calculated stability ratios are in accordance with the experimental values [6, 7].

References

[1] Holthoff, H. et al. Coagulation rate measurements of colloidal particles by simultaneous static and dynamic light scattering. Langmuir 12, 5541-5549 (1996). doi: 10.1021/la960326e

[2] Israelachvili, J. N. Intermolecular and surface forces: revised third edition. Academic press, 2011.

[3] Russel, W. B.; Saville D. A.; Schowalter, W. R. Colloidal dispersions. Cambridge University Press, 1992.

[4] Popa, Ionel, et al. "Importance of charge regulation in attractive double-layer forces between dissimilar surfaces." Physical review letters 104.22 (2010): 228301. 10.1103/PhysRevLett.104.228301.

[5] Borkovec, Michal, et al. "Investigating Forces between Charged Particles in the Presence of Oppositely Charged Polyelectrolytes with the Multi-Particle Colloidal Probe Technique." Advances in Colloid and Interface Science (2012). 10.1016/j.cis.2012.06.005.

[6] Behrens, Sven Holger, et al. "Charging and aggregation properties of carboxyl latex particles: Experiments versus DLVO theory." Langmuir 16.6 (2000): 2566-2575. 10.1021/la991154z.

[7] Sadeghpour, Amin, Istvan Szilagyi, and Michal Borkovec. "Charging and Aggregation of Positively Charged Colloidal Latex Particles in Presence of Multivalent Polycarboxylate Anions." Zeitschrift fur Physikalische Chemie (2012). 10.1524/zpch.2012.0259.

Ferroelectric materials are class of materials with interesting electrical properties. For a material to be ferroelectric it must in addition to be polar possess a spontaneous polarization, i.e. electrical polarization in the absence of an external electric field. Furthermore the direction of polarization is switchable with an external electric field [8]. When measuring polarization vs. electric field of a ferroelectric material a typical hysteretic P-E loop is observed. During this process the polarization is switched by the electric field, see Figure 3.

Figure 3. Schematic presentation of the ferroelectric loop. Polarization-Electric field (P-E) dependence for a ferroelectric material (left). Presentation of the movement of the atoms in the unit cell during loop formation (right). The example shows ABO₃ perovskite unit cell, with A atoms colored green, B atom white, and O atoms red. The images correspond to the marked states in the loop.

Ferroelectric materials can come in a diffrent forms depending on the application they can be used as single crystals, ceramics, thick films or tihin films. All ferroelectric materials are piezoelectric and they can be used as sensor, actuators, high precesion positioning devices, etc. In my PhD I was mainly focused on the sythesis of ferroelectric ceramics materials based on lead magnesium niobate. This is a class of interesting materials with superior ferroelectric and piezoelectric properties. We developed a new approach to synthesis of these materials, which smplified the process and yielded ceramics with expetional electrical properties. For details see [9-11].

References

[8] Damjanovic, D. "Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics. Rep. Prog. Phys. 61, 1267 (1998). doi: doi:10.1088/0034-4885/61/9/002

[9] Trefalt, G.; Malic, B.; Kuscer, D.; Holc, J.; Kosec, M. Synthesis of Pb(Mg_1/3Nb_2/3)O₃ by Self-Assembled Colloidal Aggregates. J. Am. Ceram. Soc. 94, 2846-2856 (2011). doi: 10.1111/j.1551-2916.2011.04443.x

[10] Trefalt, G.; Malic, B.; Holc, J.; Kosec, M. Synthesis of 0.65Pb(Mg_1/3Nb_2/3)O₃-0.35PbTiO₃ by Controlled Agglomeration of Precursor Particles. J. Am. Ceram. Soc. 95, 1858-1865 (2012). doi: 10.1111/j.1551-2916.2012.05142.x

[11] Trefalt, G.; New Synthesis Route to Pb(Mg_1/3Nb_2/3)O₃ Based Materials by Controlled Agglomeration of Reagent Particles, PhD thesis (2012). [PDF]

Computer simulations are somehow in the middle between theory and experiment. In a purely theoretical work, a model is proposed and then theoretician tires to solve it with only "paper and pencil" and of course a lot of knowledge of mathematics. In the case of computer simulations the same model can be used but the problem is solved numerically. The solution of a theory is an analytical relation, i.e. equation, which gives you the explanation how certain properties are connected. In contrast, the result of a simulation is only a numerical value, which has a statistical error like an experimental result. In this respect the simulations resemble experiments. In the next example we will show how we can use Monte Carlo (MC) simulation to calculate the acid dissociation by randomly rolling a dice.

Let's titrate an acid with a base and see how the fraction of the dissociated acid changes with pH. The acid dissociates according to the following equation:
.
The dissociation constant is deffined as:
.
From this equation we can derive an analytical expression, which tells us how the dissociation changes with pH:
.
We can also calculate the dissociation of the acid with the MC method and compare the results with the analytical expression. In the simulation we try to convert the molecule from the undissociated state (HA) to the disociated state (A^-). There is a certain transition probability for going from one to the other state, which reads as:
.
Now, we try to go from state i to state j and calculate the corresponding transition probability. We compare this probability to a randomly chosen number form the interval [0,1]. If the probability is higher than the number we accept the new state otherwise we reject it:
.
Now all we have to do is to repeat this process and count how many times we reach the state i or j. The fraction of the acid in the dissociated state is calculated as:
.
Here is a simple Octave code to calculate the fraction of dissociated acetic acid at pH = 4:

% Octave Code for MC titration

pH=4
pKa=4.76

sumj=0
j=0		%starting from state HA	

for i=1:10000	%main loop

  if(j==0)
    
      fij=exp((pH-pKa)*log(10)); %calculate the probability
				   %for going from i->j
      if(fij>rand()) 		%rolling a dice and 
	j=1;		   	   %comparing to fij
      endif
 
  else
  
      fji=exp(-(pH-pKa)*log(10)); %calculate the probability
				    %for going from j->i
      if(fji>rand())
	j=0;
      endif
  
  endif
  
  sumj=sumj+j;
  
endfor		%end of main loop

alpha=sumj/i	%fraction of disscociation is
		  %number of j states divided by
		  %number of all states
						  

In the figure below the result calculated with the above code is presented together with the analytical solution of our problem, see equations above. We can see that the two results are in a perfect agreement. And this is what the MC simulation usually gives - exact solution of the model that we are using.

Figure 3. Fraction of dissociated acetic acid as a function of pH. Comparison between analytical solution (solid line) and MC simulation (symbols).

The method actually got its name after the city on the Mediterranean coast because it uses a lot of random numbers, which is basically the same as rolling a dice - the main thing that is going on in Monte Carlo.

Now lets come to an real example from our research, where MC is used to see what is the structure of the colloidal clusters in the suspension, where three types of particles are aggregating. Suspension containing lead oxide, magnesium basic carbonate, and niobium oxide particles is used for the synthesis of lead magnesium niobate (PMN) ceramics. It turns out that the structure of the aggregates in this system is crucial for the synthesis of the phase-pure product. Therefore we are using MC simulations to predict the structure of colloidal aggregates at different experimental conditions, namely the pH.
To be able to run the simulations we first have to determine the interactions between the particles in the suspension. This can be done by employing DLVO theory based on the experimental surface potential, which we can get from the zeta-potential measurements (see Figure 4).

Figure 4. Interactions between reagent particles for the synthesis of PMN, namely PbO, MHC (magnesium hydroxy carbonate), and Nb₂O₅ at pH = 11.4 (left) and pH = 12.5 (right).

When we know the interaction potentail we can run the MC simulations and see what is the structure of the aggregates that we can expect at different experimental conditions, namely at pH = 11.4 and pH = 12.5, see Figure 5.

Figure 5. Spatial distribution of colloidal particles of PbO (red), MHC (blue), and Nb₂O₅(green) in suspensions at pH = 11.4 (left) and pH = 12.5 (right) conditions. In contrast to the pH12.5 aggregates, a lot of non-desirable PbO-Nb₂O₅ are present in the pH11.4 aggregates

In the case of the synthesis of the PMN and PMN based materials, the cruical requirement is to prevent the reaction between lead and niobium oxide particles. Therefore we want to avoid the PbO-Nb₂O₅ contacts in the reagent particle mixtures. As the MC show the preferable condtions for the desired arrangement are the higher pH conditions (i.e. pH = 12.5). When we synthesize the PMN from the pH11.4 and pH12.5 suspension, we can obsereve that the piece of pH11.4 ceramics is macroscopically non-homogeneous, while the pH12.5 ceramics is highly homogeneous, see Figure 6.

Figure 6. Sintered ceramic pieces of the pH11.4 (left) and pH12.5 (right) samples. The pH11.4 sample is non-homogeneous, while pH12.5 sample is highly homogeneous.

More importantly, if we measure the properties of pH11.4 and pH12.5 ceramics, we can see that pH12.5 samples outperform the pH11.4 samples by a big margin. For example the electro-mechanical response, i.e. the strain as a function of the electric-field, is approximately 10 times higher in the case of the pH12.5 sample as compared to the pH11.4 sample, see Figure 6.

Figure 7. Electrical properties of the pH11.4 (left) and pH12.5 (right) samples. Strain-electric field (S-E) loops. The respone of the pH11.4 is approximately 10 times lower as compared to the pH12.5 sample.

The predictions from the MC simulations enable us to choose the best conditions for the synthesis of PMN-based materials, and thus give us the posibility to enhance the properties of the functional materials [12].

References

[12] Trefalt, G.; Tadic, B.; Kosec, M. Formation of colloidal assemblies in suspensions for Pb(Mg_1/3Nb_2/3)O₃ synthesis: Monte Carlo simulation study. Soft Matter 7, 5566-5577 (2011). doi: 10.1039/C1SM05228D