Root mean squar thickness

Figure 2. Root-mean-square thickness in the plateau region of polystyrene, adsorbed on chrome ferrotype as a function of the square root of the molecular weight, in cyclohexane, 36.4°C.

Scheutjens and Fleer further calculated50 the concentration distributions of loop and tail segments, the root-mean-square thickness of the adsorbed layer, and the average numbers and lengths of trains, loops, and tails. They also computed the distributions of trains, loops, and tails from the free segment probability Pj, which may be represented by... 

The overall root-mean-square thickness trms can be represented in terms of 0 as... 

B.3.4 Root-Mean-Square Thicknesses of Loops and Tails B.3.4.1 Root-Mean-Square Thickness of Loops... 

Experimentally, the root-mean-square thicknesses of loops and tails can be measured by ellipsometry. It is thus necessary to relate them to the average numbers of segments in loops and tails for deducing the conformation of an adsorbed polymer. 

Hesselink39,40 derived relations between the root-mean-square thickness and the loop or tail size. For the case in which no intrasegment interaction exists, he derived the segment distribution p4(z) for a single loop of size i by considering all posable configurations of the chain that starts at the interface and returns at its end to the interface. His expression for p4(z) reads... 

Ellipsometry determines a certain average thickness th of the adsorbed layer. However, what is important for the evaluation of polymer conformations in this layer is the root-mean square thickness t. Hence, it is necessary to find a way of relating t to th. McCrackin and Colson66 studied this problem for several distributions of segments and found tnn, = th/l-5 for the exponential distribution and t - th/1.74 for the Gaussian distribution. Takahashi et al.67 showed that t = th/1.63 for the one-train and two-tail model (see Eqs. (B-110) and (B-lll)). 

Fig. 12. Root-mean-square thickness tms of the adsorbed polymer layer vs. polymer concentration70. Symbols are the same as in Fig. 11...

Fig. 13. Root-mean-square thickness t, of the adsorbed polymer layer at the bulk polymer concentration 0.3 g/dl vs. molecular weight70 ...

From studies performed with well characterized substrates and polystyrene with a narrow M distribution, the measured values of T, p, and 0 at the theta point have been found to agree closely with the theories of Silberberg and Scheutjens and Fleer. Furthermore, it has been shown that the measured root-mean-square thickness of the adsorbed layer can be predicted semiquantitatively by the loop-train-tail conformation model. 

Measurements of hydrodynamic thickness LH have been performed by many investigators and, in most cases, the measured LH were almost twice the radii of gyration of polymer coils in bulk solution. It is desirable to clarify the theoretical relationship between LH and the root-mean-square thickness of the adsorbed polymer layer. Some progress in this direction has been made recently. 

The arrows along the abscissa in fig. 5.19 indicate various layer thicknesses the elllpsometric thickness d . the root-mean-square thickness d , and the hydrodynamic thickness d . For a definition of these quantities we refer to sec. 5.6. From fig. 5.19 it is clear that d " and d are meilnly determined by the loops, whereas measures the extension of the tails. We return to this point below. 

The thickness of the layer may also be defined as a derived parameter fi-om the volume fraction profile c )(z) of the adsorbed polymer layer the most common feature of which is the second moment or root-mean-square thickness of the layer ... 

The root mean square thickness of the adsorption layer can be described by 1... 

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