Experiments involving polystyrene latex often require researchers to calculate physical parameters of the particles used. These parameters include particle size and area, number of particles in a given volume, and charge density and area. The information given in the sections below will assist you in the correct calculation of these parameters for your experiments.

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Particle size, D in μm

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This number average diameter, D, is given in µm. It is determined by transmission electron microscopy using a diffraction grating replica as a calibration standard. If special traceability is required, NIST particles are included as part of the calibration process. The calibration standard is photographed each time a latex sample is sized. Scanning electron microscopy is not suitable for good particle sizing as the 3-D effects result in an uncertainty in size. At least 500 particles are measured for each sample; we use a purposely built image analysis program called “Boolero” which is available for purchase. The weight average diameter is also calculated as is the standard deviation on the mean. This is given on each certificate of analysis as the coefficient of variation, cv, which is simply the standard deviation, δ, expressed as a percentage of the number average diameter:

                            
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Specific surface area, SSA in cm2g-1

This is the total surface area of all the polymer microspheres making up 1g of polymer and is given in units of cm2g-1. This number is important when carrying out any adsorption work, as this will effect the capacity of the microsphere batch for protein take-up. The SSA has also been termed the area-to-mass ratio, but specific surface area is the preferred nomenclature.

First we have to calculate the mass and surface area of one microsphere. The surface area of a sphere is:

                              
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And the volume is:

                            
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And as the mass is the volume multiplied by the density, ρp the SSA is:

                        
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Now the size is given in μm and the density in g/cm3, so we multiply the diameter by 10-4 to convert the diameter into cm:

                        
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The density of polystyrene at room temperature is 1.055 g/cm3. If other polymers are used for the microspheres their density should be used.

Latex concentration, Wv% in g/100 mL

This is given as the percentage of solid polymer. It is measured by dry weight. A aliquot of latex is weighed and then dried in an oven and the weight of dry polymer determined. This gives the weight ratio, Ww, of polymer to latex. So in 1g of latex we have:

                              
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These are readily converted to volumes using the density of the polymer, ρp, and the density of water ρw. The total volume is the volume of water + the volume of polymer:

                              
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Number of particles per mL, Np in mL-1

We have already calculated the weight of a single microsphere as the volume times the density of the polymer:

                          
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The number of particles per mL is calculated from the mass of polymer per mL divided by the mass of a single microsphere. From the solids content we have Wv%/100 g of polymer in 1 mL:

                              
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where the factor of 10-4 is to convert the diameter in μm into cm. When simplified, this gives us the expression:

                              
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Surface charge density, σ in μC/cm2

The charges on the surface of latex microspheres are key components of the colloid stability of the particles and for carboxyl groups; for example, they indicate of the maximum number of possible sites for covalent attachment of proteins. Without a significant surface charge, surfactants must be added to give the colloid stability and prevent aggregation. The surface charge of a latex microsphere can be determined by titration. Because of the small amount of charge material present, this is a demanding technique. The latex must have no free acids or bases present which are not attached to the polymer of the particles and all the bound acid or base groups must be in the H or OH form. The equivalence point of the titration is determined by the H or OH form. The equivalence point of the titration is determined conductometrically and is found as the number of micro-equivalents of base or acid used to neutralize 1 g of polymer, Et.

The surface charge density is then readily calculated by using the Faraday Constant (F = 96,485 Coulombs per equivalent) and the Specific Surface Area.

                      
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Which simplifies to

                        
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where we have taken the polystyrene density as 1.055 g/cm3.

The Area per charged group, A in Ǻ2

Firstly we have to convert the Specific Surface Area into Ǻ2/g for cm2/g. There are 108 Ǻ per cm. Then we have to divide the SSA by the number of charged groups per gram using Avogadro’s Number to convert the number of equivalents to number of charged groups:

                        
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Which gives :

                         
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Of course, this must be viewed as density, based on the geometric area of a dried particle. So, with sulfate microsphere, for example, the area per sulfate group may be 10 to 20 times the cross-sectional area of a sulfate group. This means that the surface around each sulfate is the hydrophobic polystyrene. On the other hand, with a CML particle the area per group can easily be so high that a 3-D layer of charged groups is present.