NBS

PUBLICATIONS A11104 DSS1D1

NBSIR 79-1722

Characterizing the Interfiber

Bonding of Paper Pulps:

Rationale for Bonding

Parameters Derived From

Tensile Test Data

Jack C. Smith

Polymer Science and Standards Division

Center for Materials Science

National Bureau of Standards

April 15, 1979

Progress Report Covering the Period

October 1, 1978 through December 31, 1978

Prepared for Bureau of Engraving and Printing

reasury Department

-nr Washington, D.C. 20234

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79-1722

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NBSIR 79-1722

« i • CHARACTERIZING THE INTERFIBER

BONDING OF PAPER PULPS:

RATIONALE FOR BONDING

PARAMETERS DERIVED FROM

TENSILE TEST DATA

Jack C. Smith

Polymer Science and Standards Division

Center for Materials Science

National Bureau of Standards

April 15, 1979

Progress Report Covering the Period

October 1, 1978 through December 31, 1978.

Prepared for Bureau of Engraving and Printing

Treasury Department

Washington, D.C. 20234

U S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

NBSIR 79-1722

CHARACTERIZING THE INTERFIBER

BONDING OF PAPER PULPS:

RATIONALE FOR BONDING

PARAMETERS DERIVED FROM

TENSILE TEST DATA

Jack C. Smith

Polymer Science and Standards Division

Center for Materials Science

National Bureau of Standards

April 15, 1979

Progress Report Covering the Period

October 1, 1978 through December 31, 1978.

Prepared for Bureau of Engraving and Printing

Treasury Department

Washington, D.C. 20234

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

TABLE OF CONTENTS

SUMMARY AND CONCLUSIONS . . . . . Q . . INTRODUCTION . THE TENSILE TEST . 0 . „ . # 0 0 „ . CHARACTERIZING THE TIGHTNESS OF A WEB STRUCTURE

AN ENERGY PARAMETER

FOR CHARACTERIZING BOND STRENGTH . „ . A FORCE DROP PARAMETER

FOR CHARACTERIZING BOND STRENGTH • • o DISCUSSION ............

A BOND STRENGTH PARAMETER

THAT IS INSENSITIVE TO MESH SIZE . . . REFERENCES

i

SUMMARY AND CONCLUSIONS

As a result of some recent research a new technique is

evolving for characterizing the interfiber bonding of paper

pulps. Handsheets in the form of very low density webs are

made from the pulp to be evaluated. Specimens from these

handsheets are then extended in a sensitive tensile tester to

obtain a force-elongation curve, from which various bonding

parameters are derived.

The force-elongation curve for a web specimen has a

jagged appearance; the force rises and dips through one or

more peaks, and eventually diminishes to zero. Each jag in the

curve is assumed to indicate the breaking of a bond between

fibers. If the average elongation between successive jags is

small, the web has a relatively large number of fiber crossovers per unit area, and most of these crossovers are bonded.

On the other hand, if the average percent elongation per break

is large, the web has only a few interfiber bonds per unit area.

This means that there is a light density of crossovers, or that

many of the crossovers are not bonded, or both. In order to

characterize the areal density of interfiber bonding, the

percent elongation of the web is plotted versus the number of

bond breaks, and the initial slope of this curve is used as the

appropriate parameter.

Each time a bond breaks as the network is being extended,

the force drops abruptly, indicating that some of the fibers

are no longer carrying load. The magnitude of a force drop is

related to the force that one of the interfiber bonds was able

to withstand before it gave way. For example, if the bond is

strong, the fibers connected by it carry a heavy load, and the

force drop resulting from bond break is large. Thus the average

of a series of force drops should provide a parameter for

characterizing bond strength. Unfortunately the magnitude of a

force drop also depends on the size of the mesh that is opened

up by the break. As there is a wide variation of mesh sizes
ii

in a web network, there is a correspondingly wide variation

of force drops. Thus it is difficult to determine an average

force drop that adequately gauges interfiber bond strength.

Each time a bond breaks, some of the strain energy stored

in the network is dissipated. This energy loss can be calculated from the slope of the force-elongation curve just before

a break occurs, and the values of the force just before and

just after break. The energy loss is closely related to the

strength of the bond that breaks, so that the average of a

series of energy losses should provide a parameter for characterizing bond strength. Unfortunately the value of the

energy loss parameter is dependent on mesh size and mesh size

distribution, thus its effectiveness in gauging bond strength

is limited.

Studies with mathematical models simulating a network of

pulp fibers have suggested that a force drop is approximately

0 5 proportional to f , and the corresponding energy loss is

approximately proportional to f * , where f is the force

before break in the fibers joined by the bond. The studies also

suggest that the force drop and energy loss are approximately

linearly proportional to a length characterizing the mesh

involved in the bond break. Thus the ratio of energy loss to

force drop should be dependent on bond strength but insensitive

to mesh size. A good bond strength parameter could be obtained

by averaging this ratio over a series of bond breaks. This

parameter measures the energy loss per unit force drop. It has

the dimensions of length. Thus it provides a characteristic

elongation representative of the amount a network must be

stretched in order for a bond to break.

This report comprises a consolidation of knowledge derived

from previous researches. It suggests several parameters for

use in characterizing interfiber adhesion. These parameters

must be tested experimentally . Tests should also be made to

verify the assumptions involved in the derivation of these

parameters and in the interpretation of their meaning.
1

INTRODUCTION

There is a need to develop parameters that can be used

to characterize quantitatively the adhesion between fibers in

a sheet of paper. Such parameters, for instance, would be

useful in selecting, blending and treating batches of pulp to

produce a paper with improved interfiber adhesion.

One way to obtain information on interfiber adhesion is

to perform tensile tests on handsheets in the form of lowdensity open webs made from the pulp of interest. Appropriate

parameters can be derived from the force-elongation curves

obtained in these tests. Bond strength can be characterized

in terms of an energy parameter, obtained by averaging the

energy dissipated by the fibrous network for each of a number

of selected bond breaks. Alternative characterizations can be

obtained by averaging the drops in force sustained by the

network for each of a number of bond breaks, or by calculating

a characteristic elongation representative of the amount the

network must be stretched in order to cause a bond break.

Another useful parameter is obtained by plotting the

percent elongation of the specimen as a function of the number

of bond breaks. The initial slope of this curve, or initial

percent elongation per break, can be used as a parameter

characterizing the areal density of bonding between fibers

constituting the network.

In order to derive these and other parameters, and to

interpret their meaning, it is necessary to make some assumptions about the mechanical processes taking place as the

fibrous network is elongated. It is also necessary to have some

ideas about the effect of such variables as network size,

specimen shape and existence of flaws. These topics have been

discussed in previous reports [1-4], and a better understanding

of them has gradually developed. In this report an attempt is

made to consolidate these and other concepts, and discuss them

in depth in order to record the current understanding and

provide a background for future research.
2

THE TENSILE TEST

Consider a random fibrous network formed from the pulp

to be evaluated . The network should be an open web approximating a "2-dimensional" web in thinness . In a 2-dimensional

web the number of constituting fibers is so small that the

area covered by more than two fibers intersecting at a given

point is negligible. The concept and statistical geometry

of 2-dimensional networks has been discussed by Kalmes and

Corte [ 5 ]

If a specimen cut from this network is stretched to break

in a sensitive tensile tester, a jagged force-elongation curve

is obtained in which the force rises and dips through one or

more peaks, and eventually diminishes to zero. Each jag in

the curve signifies the breaking of a bond between fibers.

(it is assumed that the fibers are much stronger than the

bonds between them so that the fibers themselves do not break)

During the straining operation the following behavior

sequence is observed in the fibrous network: The fibers tend

to align themselves in the direction of the extension with

the straightest and most perfectly aligned fiber segments

bearing most of the load. As the specimen is elongated it

contracts laterally forming a fluted structure in the portions

not restrained by the end clamps. Bond breaks often occur at

random points, causing only local damage and leaving the

surrounding network essentially intact. However a breakage

of some bonds usually in the vicinity of network flaws or

irregularities, results in more extensive damage. Often a

succession of breaks occurs in the same region because of

stress concentrations that develop. In the regions of more

extensive damage parts of the network lose their strain and

sag away. As the extension continues further deterioration

occurs. Portions of the network midway between the ends of

the specimen tend to collapse into threadlike structures in

which the longitudinal forces are concentrated. These forces
3

in turn are diffused through the uncollapsed portions of the

network at either end. In the final stages of the breaking

process often only one long threadlike structure remains.

This structure sustains relatively large elongations before

breaking

Figure 1 is a tracing of the force-elongation curve

2 obtained in an actual test on a well bonded web of 2.5 g/m

mass per unit area. The specimen, 2 cm long by 1 cm wide, was

elongated at a rate of 0.2 cm/min. Chart speed was 20 cm/min,

and full scale deflection corresponded to a force of 50 mN

(5 g) ° (Note that the elongation scale on this tracing reads

from right to left). The points at which bonds are thought to

break have been indicated and numbered.

Various types of jags in the force-elongation curve are

encountered throughout the course of the test. The jags

numbered 1, 3> 12, 19 * 3

0

> 57 are all well defined, and the

associated drops in tension are small. These jags are thought

to denote bond breaks in which only local damage results.

At points 4, 15* 26, 44 , 47 the force exerted by the

network does not drop, but over a short elongation remains

constant or else does not increase at its previous rate. This

behavior may result from the breaking of weak attachments

between fibers that were only slightly stressed, because of

low orientation or because of a local concentration of

neighboring fibers sharing the stress. It could be due to the

action of frictional forces between fibers. Small force drops

may occur but might not be resolved if the full-scale force

setting of the recorder chart is too high. In addition, inertia

of the clamp and coupling to the load cell can slightly limit

the sensitivity to small changes in force. Although the cause

of these discontinuities in slope is uncertain, the most

noticeable ones should probably be counted as bond breaks.

At jags numbered 2, 5 * 22, 39 large drops in force are

apparent, implying bond breaks in which parts of the network
4

incur appreciable damage. However the network, although

damaged, may develop large and occasionally even greater

resistant forces as the elongation proceeds. This behavior is

exemplified by the force-elongation curve of figure 2, which

was obtained by an identical test on another specimen of the

material used in figure 1. Note here that after the force drop

at jag 27, the force level gradually rises with increased

elongation, and at jag 70 has regained its earlier value.

So far the elongation of the web has been described from

the standpoint of an observer watching the process. Two

examples illustrating the force-elongation behavior were then

given. It should be noted that most of the test specimens have

irregularities in structure, and that the size of these

irregularities is of the same magnitude as that of the specimen

itself. Thus the behavior of any given web specimen may be

different from that of another specimen of the same material.

The description given only points out behavioral features that

may or may not occur in each instance. In order to understand

the behavior, an assumed network response mechanics is now

presented. Some of the expected behavior of the web network is

described, but the extent to which features of the behavior

take place will vary from specimen to specimen.

In the early stages of the stretching process segments of

the fibers composing the network tend to orient themselves in

the direction of the extension, and to resist the extension

through the combined action of forces along their length. As

the extension proceeds more and more of these segments between

bonds become oriented and bear load. As a result of this

process the force-extension curve has a slope that increases

with increasing elongation. This initial nonlinear region is

apparent in the force-elongation curves of figures 1 and 2.

After this initial behavior a state of elongation is

attained in which all of the segments that can be oriented now

share in maintaining the force resisting stretch, and the
5

force-elongation curve becomes more linear. The total force

however is still sustained by relatively few fiber segments,

and the individual forces in the segments are very unevenly

distributed. Most of the fiber segments in the network bear

little or no load, because they are unaligned, or because most

of the load is borne by adjacent fibers with straighter

segments between bonds. When the force on a given segment

becomes sufficiently great a bond at the end of the segment

breaks, and the network opens up in the vicinity of the break.

The subsequent force-elongation behavior of the network

will depend upon the size of the hole that opens up. Often the

first breaks occur randomly throughout the network, and only

small or moderate holes result leaving the network essentially

intact. In this instance other fiber segments in the vicinity

of the break assume the load relinquished by the broken bond.

In the web there is a redundancy of these previously unoriented

unloaded fibers. Thus the web can be expected to behave as

though the force overall were sustained by the same number of

load bearing elements before and after the break.

The breaking of a bond tends to increase the initial

unloaded length of the specimen. Thus the force-elongation

curve is shifted so that a given force corresponds to a slightly

greater elongation. If the bond break leaves the specimen

essentially intact, the shape of the force-elongation curve

should be almost unaffected by the break. In particular, the

slope of the force-elongation curve at the force level of the

break should be practically the same before and after the break.

This is exemplified by the first 35 breaks in the forceelongation curve of figure 1, where most of the slopes before

and after break have the same value.

The process just described in which bonds break randomly

but leave the network essentially intact changes the distribution of forces in the load bearing segments. The bonds,

bearing a disproportionate share of the load are most likely
6

to break. Thus the distribution of forces becomes more even

after a number of breaks, and better cooperation between load

bearing segments ensues. Under these conditions the network

gradually sustains a higher force level, and the shape of the

force-elongation curve changes so that the slope at a given

force level becomes greater . This effect is particularly

pronounced in the force-elongation curve of figure 2, where

the slopes in the vicinity of break 25 are greater than the

earlier slopes in the vicinity of break 5=

During the elongation of a fiber network specimen any

number of different bond breaking situations might arise.

Figure 5 depicts some of them schematically. In figure 5a, for

instance, a region of the specimen is shown in which the meshes

are uniform and of moderate size. If the bond denoted by the

black dot should break, the mesh would open up slightly, and

some of the strain energy stored in the vicinity of the bond

would be dissipated. The force with which the network resists

elongation, however, is sustained by a relatively large number

of parallel fibers in the surrounding meshes, so that the

decrease in force resulting from the break would be small. The

force-elongation behavior in this instance might be somewhat

analogous to that of a number of short springs in parallel.

In the situation depicted in figure 5b, two bonded fibers

are holding together a large mesh. If the bond between these

fibers should break, a hole would form associated with a large

distortion of the network. Some of the force relinquished by

the broken bond would be assumed by the meshes to either side

of the hole, but the forces previously channeled through the

meshes at the top and bottom would be greatly diminished. Thus

a large decrease in force would occur, and a large amount of

strain energy would be dissipated.

Long strip or threadlike configurations frequently form

in the specimen after a substantial elongation. Figure 5c

represents schematically the unstrained structure of these
7

portions of the network . This striplike structure is often

observed by itself, but with increasing elongation it condenses

so that its shape is that of a single thread. The fibers in

such a long narrow condensed strip become highly oriented when

straining takes place, and the force borne by the resulting

thread is sustained by only a few parallel pathways. When a

bond breaks in the thread the force usually drops by a significant fraction, and a comparatively large elongation is usually

required before the force builds up to near its pre-break

value. The force-elongation behavior of the thread might be

considered as somewhat analogous to that of a long spring

composed of many shorter springs joined end to end.

Figure 3d depicts the appearance of a web specimen in

which three threadlike structures have formed. The' total force

borne by the specimen is concentrated in these three threads,

but elsewhere is diffused throughout the specimen. When the web

specimen has a configuration similar to this, any of the three

types of bond breaks just described may occur. For instance if

the center thread should give way the specimen would incur

significant damage as in situation Jb. The central portion of

the web would be unloaded and the large amount of strain energy

previously stored in these portions would be dissipated. The

total force would drop by a large amount, and consist of the

forces channeled through the two remaining threads. In addition

some fiber segments in the web portions at the ends of the

specimen might lose their previous orientation and carry less

load. With increasing elongation the force level would build

up, but the force-elongation curve would have an initial region

in which the slope increases with elongatidn as the load

bearing fiber segments become better oriented. This post break

behavior can be observed in many of the later breaks of figure 1.

If in figure 3d the two threads at the sides should break

the force resisting elongation would be concentrated in the

central condensed thread, and would be diffused through the
8

uncollapsed parts of the network at either end. If the

parallel fiber segments constituting the central thread are

well oriented, and the forces in these segments are evenly

distributed, the total force sustained by the thread may be

quite large. In such case the force level may rise to high

values, and the bond breaks will most likely occur in the web

portions of the specimen at either end. The appearance of the

force-elongation curve will be similar to that for a specimen

that remains intact during a series of breaks.

If the forces in the thread are not evenly distributed,

and are concentrated in only a few parallel pathways, the

thread will be unable to sustain a high force level. With each

successive bond break in the central thread, this force level

will diminish. Furthermore the slopes of the force-elongation

curve between successive breaks will be low and nonlinear,

because many of the load bearing fibers in the uncollapsed end

portions of the specimen will have lost most of their previous

orientation. Post-break behavior at low force levels in which

the slopes are low and nonlinear can be noted in the concluding

breaks of the curves in figures 1 and 2.

Some of the behavior just described is further exemplified

in the force-elongation curves of figures 4 and 5« These curves

were obtained from tests on specimens of the same material used

in figure 1. The specimens were 2 cm long and 0.5 cm wide, but

in other respects the test conditions were the same as before.

During the elongation the condition of the specimen was noted

on the chart, so that the effects of progressive damage to the

specimen could be ascertained.

In figure 4 the post break slope increases for breaks up

to the 10th, indicating a severe maldistribution of forces

among the load bearing fiber segments. The specimen web remains

essentially intact up to the 37th break when a strip forms.

Post-break slopes up to the 45th break are low, but increase

as the load distribution improves in the fibers composing the
9

strip. The strip condenses into a thread at break 45. The

force rises to a relatively high value for breaks 52 through

55# which probably take place in the web rather than in the

thread. The force then drops almost to zero, but rises again

to a high value at the final break.

In figure 5 the specimen web remains essentially intact

up to break 54 when a gap opens on the right hand side. Despite

the web damage the slope of the force-elongation curve is only

slightly decreased for breaks up to break 45 when the force

drops and a strip starts to form. The force then rises after

a series of breaks to break 60 when the strip condenses into

a thread. Post-break slopes from break 55 to break 70 have

almost the same high value, indicating that most of the breaks

are occurring in the web rather than in the strip or thread.

The final post break slopes from break 78 on have a low value

indicating that breaks are occurring within the thread.
10

CHARACTERIZING THE TIGHTNESS OF A WEB STRUCTURE

A tightly bonded web structure is one that has a large

number of interfiber bonds per unit area. The number of these

bonds depends upon the number of fiber crossings and on the

number of crossings that bond together. This concept is

illustrated in figure 6. Here the symbol A denotes a web in

which the meshes are approximately uniform in size and bonds

occur at each fiber crossover. Symbol B represents a web

having the same number of fiber crossovers per unit area as

A, but in which only those crossings denoted by dots are

bonded. The areal density of bonding of web B is less than

that of A, so web B is said to be relatively looser. Web C

is a more finely meshed web with greater areal density of

bonding than web A, and so is said to be relatively tighter.

If a structure is tightly bonded it can be seen intuitively that on the average only a small percent extension

is required before the force applied to one of the bonds is

sufficient to break it. For a loosely bonded network a larger

percent extension on the average is required. Thus a small,

initial value of average percent extension per break would

be characteristic of a tight network, and a larger initial

value of average percent extension per break would be

characteristic of a looser network. The initial average

percent extension per break for a given web specimen is equal

to the Initial slope of the curve of percent extension versus

number of breaks obtained by a tensile test.

The areal densities of bonding of two different networks

thus can be determined relative to each other by comparing

data from tests on specimens of each network. In order to

insure a reliable comparison however, the specimens used should

all have the same dimensions, and the comparisons should be

made between average values from a number of tests. It is

also desirable that the areal density of fibers in the two

networks should not be too large; otherwise a high level of
11

force would prevail during extension, and the sensitivity of

the test record to small changes in force would be correspondingly reduced. In such case some of the breaks would not be

detected.

An example of a plot of percent elongation versus number

of bond breaks for a typical web is given in figure J. Data

for this plot were obtained from the force-elongation curve,

figure 1. The initial portion of the curve, figure 7, is

linear up to break 50 » The value of the initial slope,

determined by a least squares fit of the elongation values

between breaks 5 and 50, is 0.149 % per break. In the later

stages of elongation the average extension between breaks

increases, reflecting the increased damage to the web and

the formation of strips and threadlike structures. Other

examples of elongation-break curves can be found in previous

reports [1-5]
12

AN ENERGY PARAMETER FOR CHARACTERIZING

BOND STRENGTH

In the discussion of figure 1 the jags numbered 1, 3,

12, 19, 30 , 57 were selected as examples of well defined bond

breaks with small associated drops in force. Presumably only

local damage results from these breaks. Drops in force

associated with some of the other breaks are much larger,

reflecting more extensive damage to the network. Some force

drops are so small as to be almost undetectable on the record

chart. However a range of breaks in which only local damage

occurs could be defined by requiring that the associated force

drops lie between two suitably chosen values. If the energy

dissipated by each break in this range is calculated, an

average energy per break Eq could be determined for the

specimen. The E

Q

s for a number of specimens could then be

averaged to obtain a value E that could be used as a parameter

to characterize bond strength. For instance, if the fiber

crossings in the web specimens are strongly bonded, this energy

dissipation parameter would have a high value, and if the

crossings are weakly bonded the value of the parameter would

be low.

In order to estimate the average energy dissipated per

bond break it is necessary to make some assumptions about the

relationship between the force sustained by a network and the

energy stored in it. In an extension process in which none of

the interfiber bonds are broken this relationship could be

found by integrating under the force-elongation curve to

obtain data on work of extension versus force. During the

extension a small part of the work done would be dissipated

by friction involved in aligning fiber segments, but if the

extension were removed and the specimen then reextended, most

of the frictional dissipation would not recur, and the work

of extension would be stored as strain energy in the network.
15

Thus a force-energy relationship would be obtained. It is

important to know how a force-energy relationship of this

type is affected by a bond break.

According to the previous discussion the shape of the

force-elongation curve should be almost unaffected when a bond

breaks, if the network remains essentially intact. This can be

checked by noting whether the slope of the curve is the same

before and after the break. The principal result of the break

then is to shift the force-elongation curve so that a given

force corresponds to a slightly greater elongation. In this

circumstance the force-energy relationship for the network is

unchanged by the break; i.e., points on the force-elongation

curve before and after the break, where the force is the same,

correspond to states of the network in which the stored energy

is the same. The calculation of energy dissipated by a bond

break depends on the validity of this assumption.

Figure 8a depicts a jag of the type suitable for use in

calculating the energy dissipation. On this diagram point 0

designates the force and elongation values prevailing just

after a bond break. As a specimen is elongated the force

increases up to its value at point 1 where the next break occurs,

and the force drops to point 2. During subsequent elongation

the force should rise again in such a way that the slope of the

force-elongation curve is the same as it was before the break.

Elongation e^ is the elongation at which a force is attained

that is equal to the force after break Fg. Presumably the

energy stored in the network at elongation e^ is the same as the

energy remaining in the network after the break. The work done

in extending the network from elongation e^ up to elongation

e-^ is therefore all dissipated when the bond breaks. Its value

U is given by

F

1 + F

2

( e

i e

^) u =

2

( 1 )
14

In some instances such as that depicted in figure 8b

the energy dissipated when a bond breaks exceeds the work

done in extending the network between breaks . However by

performing the construction indicated, eq (l) can still be

used to calculate the energy dissipated., In this case the

accuracy of the calculation can be improved by estimating

the slope between points O' and 1, using other slopes in the

vicinity as a guide and drawing the construction line with

this slope.

Sometimes the force-elongation curve immediately after

break is not linear, but becomes linear just before the next

break. In such case it may be necessary to use a slope

partially estimated from neighboring values, as in figure 8c,

before calculating the dissipated energy.

If the force drop F^ - F

2 is designated by F

d

, and M

is the slope of the force-elongation curve between points

0 (or O’) and 1, the difference in elongation e-^ - e^ is

equal to F^/M. Eq (l) then becomes

U =

F F, av d

M

( 2 )

where FOTr is the average force (F-, + FQ )/2. This alternative

calculation provided by eq (2) is more convenient because

the construction for determining e^ is unnecessary.

The energy parameter Eq for a given web specimen is

obtained by averaging the calculated energy dissipations U

associated with a selected group of jags on the forceelongation curve. The jags are chosen so that the force drop

F^ lies between two bounding values and D

u

. The lower bound

D eliminates from consideration those jags that are not well

defined, and for which there is some uncertainty as to the

nature of the associated bond break. The upper bound D

u

screens

out bond breaks that result in significant damage.
15

An appropriate value for Do might be 0.002 S, where S

is the full-scale force of the record chart . The quantity

0.002 S then would be the smallest force difference that

could be read with certainty from the chart. The value’ of

should be the same for all of the calculations. Thus if the

web specimens to be compared were tested at different force

levels, the value of to be used should be defined in terms

of the highest value of S that was used in the tests. The

values of D

u

should be selected after data from all of the

tests have been examined. This same value of D should then

be used in the calculation of Eq for each of the specimens.

In calculating Eq it is advisable to use only the data

for breaks within a certain range. The first few breaks should

be excluded because the network forces are unevenly distributed

and the load bearing segments are not well oriented at this

stage. The last 60 % of the breaks might also be excluded

because the specimen has probably by then incurred significant

damage. Although the values of U calculated from these later

breaks might not be affected by the damage, it would be prudent

not to rely on them. In a few instances the slope of the

force-elongation curve after a bond break may differ noticeably

from the slope before the break. In this event eqs (l) and (2)

are invalid, and the break involved should be excluded from

the calculation of Eq.
16

A FORCE DROP PARAMETER FOR CHARACTERIZING

BOND STRENGTH

Adhesion between pulp fibers can be characterized

alternatively by means of a force drop parameter F. This

parameter can be obtained by averaging force drops incurred

in a series of bond breaks to obtain Fq for a given specimen,

and then averaging the FqS for a number of specimens. The

parameter F has not been tested experimentally but its use as

a parameter to characterize adhesion seems feasible.

In order to calculate an energy parameter E it is

necessary to measure the slope of the force-elongation curve

in the vicinity of a force drop. The parameter F can be

calculated more simply, as only the force drops are used

in its evaluation.

For a definition of Fq consistent with the procedure for

obtaining Eq, the first few force drops and the last 60 % of

the force drops should be excluded from the average. The

values of the force drops used in the average should be

greater than a lower limit D^ and less than an upper limit D

u

The quantities Dg and D

u

should be the same as those used in

the calculation of the energy parameter E. In some instances

the slope of the force-elongation curve after a force drop

may differ noticeably from the slope before the drop. When

this occurs the force drop involved should be excluded.

In order that the force drop parameters Fq be mutually

comparable the test specimens must all have the same initial

length. In other respects the force drop parameter F is

similar to the energy parameter E. Both parameters are

sensitive to mesh size and mesh size distribution in the test

specimen network. Model studies [4] suggest that neither

parameter is linearly related to the average of the local

bond breaking force, and that the functional dependence of

F on the average bond breaking force is different from that

of E.
17

DISCUSSION

The force-elongation curve of a web network specimen

provides various data that may be of use in estimating interfiber bonding forces ; such as, forces sustained by the

specimen, elongation to break of the specimen, slopes of the

curve, average elongation between bond breaks, and the force

drops resulting from bond breaks . The force drop data are most

directly related to the individual bond breaks; thus an

average force drop parameter such as F would apparently provide

a good characterization of interfiber adhesion. An energy

dissipation parameter such as E, which is derived from force

drop data and the slopes of the curve, should also provide a

good characterization of adhesion.

Other parameters might be used; such as, the work to break

the specimen, or the average force level obtained by dividing

the work to break by the elongation at break. These and similar

parameters would seem to be more practical because they are

easily measured. However, they are likely to vary appreciably

from specimen to specimen, because they are very sensitive to

structural imperfections. The parameters E and F derived from

force drop data and slopes are not so sensitive in this respect.

The parameters E and F are also better understood; i.e., it is

possible to describe in a qualitative way what kind of average

interfiber bond strength these parameters represent.

Model studies in which a fibrous network was simulated by

arrangements of springs in parallel [4], suggest that when a

bond breaks, the energy lost by the network is approximately

independent of the size and length-to-width ratio of the

specimen that is being elongated. To a similar approximation

the force drop is inversely proportional to the length

of the specimen, but is independent of its width. If these

assumptions are valid for actual test specimens, one might

infer that the values of the force drops and energy losses
observed in a test are not appreciably affected by cumulative

damage to the specimen as it is being elongated. This is an

important advantage

.

The force drop and energy loss parameters also have

serious disadvantages:

1. The force drops and curve slopes cannot be measured very

precisely.

2. The model studies suggest that the force drop resulting

from a bond break is approximately proportional to a length

characteristic of the hole that is opened up. As there is a

large variation of mesh sizes in a test specimen, the deviations of the individual force drops and energy losses from

their average values will be large.

3. Some experimental studies have shown that the frequency

distributions of the force drops and energy losses are not

level; i.e., there are many small force drops and energy losses,

and the frequency of the drops and losses diminishes as their

size increases. As more information becomes available on these

frequency distributions, it may be possible to derive improved

parameters that take the distributions into account. This,

however, has not yet been done.

4. Model studies [4] suggest that the force drops and energy

losses are not directly proportional to the local forces in

the fiber segments separated when a bond breaks.

5- Bond breaks sometimes occur between fiber segments that are

not aligned close to the direction of stretch.

6. The magnitude of a force drop or bond break thus depends

in a complicated way on the mesh size, the mesh size distribution, the forces in the fiber segments involved in a bond

break, and on the orientation of these fiber segments. These

individual effects probably cannot be separated out, but ways

might be found to minimize some of the undesirable ones.

7. In order to characterize the adhesive properties of different kinds of pulp fibers, using the E and P parameters, it
19

is necessary that the web specimens formed from these pulps

have the same mesh size and mesh size distribution.

In order to obtain good precision in the measurement of

E and F, the force drops should be as large as possible.

Therefore the length of the specimen should be small. Practical

considerations require that the length of the specimen should

be at least twice the width, in order to minimize the number

of load bearing segments appreciably inclined to the direction

of stretcho The width of the specimen should be small so that

the full scale force setting on the recorder chart can be made

small, and a small change in force will then cause a noticeable

jag on the force-elongation curve. The areal density of the

sample from which the specimen is cut should be as low as

feasible, so that the average mesh size of the specimen will

be large.

The model studies [4] suggest that the force drop resulting

Q from a bond break is approximately proportional to f , where

f is the force at which a bond at one end of a load bearing

segment breaks, and a is a constant. For a model consisting

of equal square meshes formed from springs with the same

spring constant, a is approximately 0.5 . The model studies

also suggest that the energy loss resulting from a bond break

o is approximately proportional to f, where |3 is approximately

1.5 . These differing values for a and |3 imply that the force

drop parameter F is not as sensitive to the interfiber bonding

force f as is the energy loss parameter E.

The value of Eq obtained from the data for a test specimen

may perhaps be interpreted in terms of the following formulation : Let the energy losses be measured for each of a series

t h of n breaks, and let the energy loss IL for the i break be

given by

. u

± = Kf(e1 ,f

1 )g(x1

).

K is a constant that may vary slightly from break to break.
20

if the network is significantly altered by the breaks. g(x)

represents the functional dependence of on a characteristic

mesh length x. associated with the bond i. This relationship

probably is almost a linear one. f is a function of 0., the

angle before break at which the debonded segment was oriented

with respect to the direction of stretch, and f^ the force in

the segment at bond break.

Under these assumptions the characteristic energy Eq for

the specimen is given by

li=n

E

o - &«o1 .f

1 )8(*1

) 1=1

D)

The force drop parameter Fq may be interpreted similarly. In

this instance, however, the function will be different.

It should be noted that bond breaks are most likely to

occur when 9 is small; i.e., when the connected segments are

oriented close to the direction of stretch. When 9 is large

the longitudinal force along a fiber segment is small, and

debonding would only occur if the bond was weaker than average.

Thus the behavior of a test specimen might be approximated by

that of a model consisting of springs arranged to form square

meshes, and for which 9 is zero. For such a model the function

f used in the calculation of E

Q is approximated by f^ , and

the function f used in the calculation of Fn is approximated

by
21

A BOND STRENGTH PARAMETER THAT IS INSENSITIVE

TO MESH SIZE

The most serious shortcoming of the E and F parameters

is their dependence on the mesh sizes in the network that is

being tested. There is a large variation of mesh sizes, so

standard deviations of the values obtained for Eq and Fq from

measurements on a given specimen will be large. It seems

reasonable to assume, however, that for a given bond break the

dependence on mesh size will be almost the same for the energy

dissipated U, and for the force drop F^. Thus the ratio U/F^

should be almost independent of mesh size. This suggests the

possible use of a new parameter G, obtained by averaging values

of U/F^ for a given specimen to obtain a value Gq, and in turn

averaging the G

q

s for a number of specimens to obtain G.

The quantity U/F^ is the energy dissipated by the network

per unit force drop; i.e., the energy associated with a mesh

of such size that a bond break would cause a unit force drop.

From eq (2) it is seen that the quantity U/F^ is equal to

F /M, and thus has the dimensions of length. The average of

av'

the quantities U/F^ obtained from a number of bond breaks, then,

would give a characteristic elongation; i.e., a length characteristic of the elongation that must be given a network in

order to produce a bond break.

The quantity F /M is depicted graphically in figure 9 > av

where the force-elongation curve for a network is shown

schematically. Here BC is the force drop F^, and CE is the

average force F_ TT . The line AB is constructed tangent to the

av

curve at B and thus has the slope M. The lind CF is constructed

parallel to the line AB, and intersects the elongation axis at

point F. The length FE thus represents the elongation F /M.

This elongation is seen to be the amount the network must be

extended to attain a force F_„, assuming the force-elongation

curve to be linear with a slope M.
22

The parameter G may be considered as the average energy

dissipated per unit force drop, or as a characteristic length

related to the average elongation required to produce a bond

break o As the parameter G is assumed to be insensitive to

mesh size of the network for which the bond strength is being

characterized, it should be especially useful in evaluating

the relative bond strengths in networks having different areal

densities. Thus bond strengths between fibers of different

materials or fibers of different morphology could be compared.

This parameter, however, has not yet been tested experimentally,

so its usefulness has not been ascertained.
25

REFERENCES

lc Jo Co Smith and E. L» Graminski, Characterizing the InterFiber Bond Strengths of Paper Pulps in Terms of a Breaking

Energy, NBSIR 76 - 1148 » Available from National Technical

Information Service, PB 264, 689

.

2. Jo Co Smith and E. L. Graminski, Characterizing the InterFiber Bond Strength of Paper Pulps in Terms of a Breaking

Energy: Effect of Beating, NBSIR 77 - 1286, Available from

National Technical Information Service, PB 276, 475

5. Jo C. Smith and E. L. Graminski, Characterizing the InterFiber Bonding of Paper Pulps: Effect of Preparation Pressure

on Tensile Test Specimens, NBSIR 78 - 1459* Available from

National Technical Information Service, PB 280,279*

4. J. C« Smith, Characterizing the Interfiber Bonding of Paper

Pulps: Tensile Behavior of Some Mathematical Models of

Paper Networks, NBSIR 79 - 1755* Submitted for publication

in Journal of Research of the National Bureau of Standards.

5» 0. Kalmes and H, Corte, TAPPI 45 , 757 - 752 (i960); Errata,

TAPPI 44, 448 (1961)

.
,
Figure

1.

Force-elongation

curve

for

a

well

bonded

web

of

pulp

p

fibers,

2.5

g/m

mass

per

unit

area.
o

i i 18

O O O O O o

10 <3

- ro cvl —

N* ‘ 30d0J

Figure

2»

Force-elongation

curve

obtained

by

a

repeat

of

the

test

used

for

figure

1

.
Figure 3° Possible situations encountered in a tensile

test of a fibrous network

.
— inOo

ino

cvj

in

CVJ ommmo

si- o O o O o

in ro CVJ ‘ 30 d 0d Eo

LU><

crhcc<Xo o 00s

•Ho0a

CQ00£O000 0 0O 0

bO

•H

0500 0

•H

O 0 <0

-P02 05 O 0 0>00O i—

0

•H00

-P0s 0O

•H

-P0

hQ 0 0 £ 0 0

rH

M 01 0 0 0H O -P 0O 01o 000

faO •H

Ph
o

Nw ‘ 30d0d 0

jr

P

<PO

-P

cd

<L) a0

cd P

Td0a

•H

cd -pPo 9 0 -=t" >

Sh 0 3 U o 3

faO £ -H o a

•H

-P

U 0 O

hD

a c0 p i—i 0 0 0 1 3 0O -P U w O 0

Ph P

LH0m3

hO

•H

Ph
Figure

6.

Webs

of

different;

tightness

of

structure

o

Web

A

has

uniform

well

bonded

meshes

.

Meshes

of

web

B

are

of

same

size

as

those

in

web

A,

but

are

bonded

only

at

points

indicated

by

dots.

Thus

web

B

has

a

looser

structure

than

web

A.

Web

C

has

a

large

number

of

well

bonded

meshes,

and

a

tighter

structure

than

web

A.
ELONGATION

0 25 50 75 100

NUMBER OF BOND BREAKS , n

Figure 7° Plot of percent elongation versus number of bond

breaks from force-elongation data, figure 1,

Initial slope of 0.149 % per break, indicated by

straight line, was obtained by a least squares

fit of elongation values between breaks 5 and 50 .
30d0d

Figure

8.

Types

of

breaks

encountered

in

calculation

of

energy

dissipation.
30UOJ

Figure

9*

Graphical

representation

of

the

quantity

F

av

/M.

A

bond

break

at

force

BE

causes

force

drop

BD.

Line

CE

represents

force

F

,a

v

and

M

is

slope

of

line

AB.

Line

CF

of

slope

M

cuts

axis

at

F.

Thus

F

/M

av'

equals

elongation

FE,

and

is

indicative

of

the

stretch

needed

for

a

break.
MBS-1I4A (REV. 9-78)

U.S. DEPT. OF COMM. 1. PUBLICATION OR REPORT NO. 1. Regent's Ac«o$sfp« No. BIBLIOGRAPHIC DATA NBSIR 79-1722

SHEET

4. TITLE AND SUBTITLE 5. Publication Date Characterizing the Interfiber Bonding of Paper Pulps:

Rationale for Bonding Parameters Derived From Tensile

Test Data

7. AUTHOR(S)

Jack C. Smith

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City, state, ZIP) 13. Type of Report & Period Covered

15. SUPPLEMENTARY NOTES

[~~| Document describes a computer program; SF-185, FIPS Software Summary, is attached.

16. ABSTRACT (A 200-word or leas factual summary of moat significant information. If document includes a significant bibliography or

literature survey, mention it here.) If low density open webs of paper pulp are tested in tension, the resulting

force-elongation curves have numerous force drops caused by the breakage of

interfiber bonds. Parameters characterizing the bonding are derived from these

data.

The relative number of bonds per unit area *s characterized by the average

elongation between bond breaks. The bond strength can be characterized by an average of the force drop magnitudes, by an average of the energy losses resulting

from bond breaks, or by a characteristic elongation indicative of the stretch in

the network needed to cause a break.

Methods of obtaining these parameters are given, and the rationale involved

in their derivation is carefully explained.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name;

separated by semicolons)

Adhesion of paper fibers; bonding of paper fibers; paper; paper fibers; bonding:

paper pulps, characterization; paper, tensile tests.

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